Topology — Volume I: General Topology / Point-Set Topology

It is one canonical Topology Volume I resource. General Topology, Point-Set Topology, Set Topology, Metric and Topological Spaces, Topological and Metric Spaces and Basic Topology are legitimate discovery/search aliases where course naming overlaps, but they must not create duplicate canonical pages.

No. Metric-space foundations required for topology are included, but Banach spaces, Hilbert spaces, operator theory and linear functionals belong to Functional Analysis and are not merged into this resource.

No. Quotients, manifolds and elementary gluing appear only as General Topology constructions. Fundamental groups, homotopy groups and homology belong to a separate Algebraic Topology resource.

The primary audience is BS Mathematics and related undergraduate programmes. It is also useful for ADP/Associate Degree discovery where Basic Topology or Metric and Topological Spaces is offered, legacy BSc revision, and MSc/advanced revision without claiming that every institution uses the same syllabus.

The final book has 45 chapters, followed by a final mathematical audit, one consolidated bibliography and the closing page.

The final master contains 414 theorem/result boxes, 133 formal definition boxes, 56 worked examples and 170 topic-wise unsolved exercises across 305 physical A4 pages.

View PDF and Download PDF should use the R2 object URL. Backup Copy should use the Drive URL. Discovery cards should open the canonical HTML resource page first.

No. Public label should be Backup Copy, matching the resource-action pattern used on The Math Hub.

Use accessible LaTeX/MathJax-style mathematics in HTML, for example A=A∪ Aprime, rather than screenshots or broken plain-text approximations.

No. The university crosswalk is curriculum/discovery context showing where comparable Topology courses occur; the resource must never claim to be official university notes unless that is factually authorized.

It covers Set algebra, indexed families, mappings, countability and the language used throughout Topology.. Its visible sections are Set Algebra, Families and De Morgan Laws, Functions, Images and Inverse Images, Countability and the Diagonal Argument, Cardinality Calculus and Power Sets. The complete PDF page range is 1–6.

Key definitions include Indexed family of sets, Image and inverse image, Countable set. Representative theorem/result blocks include Theorem - General De Morgan laws, Theorem - Inverse images preserve all unions and intersections, Theorem - Injective, surjective and bijective tests, Theorem - The rationals are countable, Cantor - \mathbb R is uncountable, Theorem - A finite product of countable sets is countable. The page should list only results genuinely present in the final resource and should preserve mathematical notation rather than paraphrase theorem identities away.

A useful visible mathematical anchor is A⊆ B, f^-1(bigcup_i U_i)=bigcup_i f^-1(U_i), |P(A)|>|A|.. This should render with the site's normal LaTeX/MathJax system and be accompanied by explanatory HTML text rather than embedded only in an image.

It covers Metric axioms, standard metrics, Minkowski inequality, equivalent metrics and basic constructions.. Its visible sections are Metric Axioms and Standard Examples, Minkowski Inequality and \ell^p Metrics, Equivalent and Bounded Metrics, Equivalent Metrics, Bounded Metrics and Quantitative Comparison. The complete PDF page range is 7–12.

Key definitions include Metric space, Topologically equivalent metrics, Lipschitz-equivalent metrics. Representative theorem/result blocks include Theorem - Discrete metric, Reverse triangle inequality, Minkowski inequality, Theorem - A bounded equivalent metric, Theorem - Lipschitz-equivalent metrics generate the same topology, Theorem - Quantitative equivalence preserves Cauchy sequences and completeness. The page should list only results genuinely present in the final resource and should preserve mathematical notation rather than paraphrase theorem identities away.

A useful visible mathematical anchor is d(x,z)≤ d(x,y)+d(y,z),   |x+y|_p≤ |x|_p+|y|_p.. This should render with the site's normal LaTeX/MathJax system and be accompanied by explanatory HTML text rather than embedded only in an image.

It covers Balls, spheres, bounded sets, open and closed sets, limit points, closure and distance to a set.. Its visible sections are Balls, Spheres, Diameter and Boundedness, Open Sets, Closed Sets and Distance Functions, Limit Points and Closure in Metric Spaces, Distance to a Set, Closure and Diameter Estimates. The complete PDF page range is 13–17.

Key definitions include Open and closed balls, Diameter, Open and closed set in a metric space, Limit point, Distance from a point to a set, Diameter. Representative theorem/result blocks include Theorem - Every open ball is open, Theorem - Distance to a set is 1-Lipschitz, Corollary - Closed sets are zero sets of distance, Theorem - Closure by distance, Theorem - Closure equals set plus derived set, Theorem - Distance to a set is 1-Lipschitz. The page should list only results genuinely present in the final resource and should preserve mathematical notation rather than paraphrase theorem identities away.

A useful visible mathematical anchor is B(a,r)={x:d(x,a)<r},   d(x,A)=inf_{a∈ A}d(x,a),   A=A∪ A'.. This should render with the site's normal LaTeX/MathJax system and be accompanied by explanatory HTML text rather than embedded only in an image.

It covers Convergence, Cauchy sequences, complete spaces, nested closed sets and Cantor intersection principles.. Its visible sections are Convergent Sequences and Uniqueness of Limits, Cauchy Sequences and Complete Metric Spaces, Cantor Intersection Theorem, Cantor Intersection Principles and Characterization of Completeness. The complete PDF page range is 18–23.

Key definitions include Metric convergence, Cauchy sequence, Complete metric space. Representative theorem/result blocks include Theorem - Limits are unique, Theorem - Every convergent sequence is bounded, Theorem - Convergent implies Cauchy, Theorem - Closed subspaces of complete spaces are complete, Cantor Intersection Theorem, Cantor Intersection Theorem - Metric form. The page should list only results genuinely present in the final resource and should preserve mathematical notation rather than paraphrase theorem identities away.

A useful visible mathematical anchor is x_n→ x ⇔ (∀ϵ>0)(∃ N)(n≥ N⇒ d(x_n,x)<ϵ).. This should render with the site's normal LaTeX/MathJax system and be accompanied by explanatory HTML text rather than embedded only in an image.

It covers Epsilon-delta continuity, sequential criteria, uniform continuity and Lipschitz maps.. Its visible sections are Epsilon-Delta Continuity, Sequential Criterion for Continuity, Uniform Continuity and Lipschitz Maps, Compactness, Uniform Continuity and Extension of Cauchy Control. The complete PDF page range is 24–29.

Key definitions include Continuity at a point, Uniform continuity. Representative theorem/result blocks include Theorem - Composition of continuous maps, Sequential criterion, Theorem - Lipschitz implies uniform continuity, Theorem - Uniformly continuous maps preserve Cauchy sequences, Heine-Cantor Theorem, Theorem - Uniformly continuous maps preserve Cauchy sequences. The page should list only results genuinely present in the final resource and should preserve mathematical notation rather than paraphrase theorem identities away.

A useful visible mathematical anchor is (∀ϵ>0)(∃δ>0):d_X(x,y)<δ⇒ d_Y(f(x),f(y))<ϵ.. This should render with the site's normal LaTeX/MathJax system and be accompanied by explanatory HTML text rather than embedded only in an image.

It covers Topology axioms, discrete, indiscrete, cofinite, cocountable and metric-generated topologies.. Its visible sections are Topology Axioms and Elementary Examples, Cofinite and Cocountable Topologies, Metric Topology and Metrizability, Generation and Comparison of Topologies. The complete PDF page range is 30–34.

Key definitions include Topological space, Cofinite topology, Metrizable space, Finer and coarser topologies. Representative theorem/result blocks include Example - Discrete and indiscrete topologies, Theorem - The cofinite family is a topology, Theorem - Open sets of a metric form a topology, Theorem - Arbitrary intersections of topologies are topologies, Theorem - Topology generated by a prescribed family, Identity-map test for comparison. The page should list only results genuinely present in the final resource and should preserve mathematical notation rather than paraphrase theorem identities away.

A useful visible mathematical anchor is ∅,X∈τ, bigcup_{i∈ I}U_i∈τ, U₁∩⋯∩ U_n∈τ.. This should render with the site's normal LaTeX/MathJax system and be accompanied by explanatory HTML text rather than embedded only in an image.

It covers Set operators, neighborhoods, boundary identities and algebraic laws of closure and interior.. Its visible sections are Interior and Exterior, Closure and Its Algebra, Boundary and Decomposition of Space, Kuratowski Closure Axioms and Duality. The complete PDF page range is 35–39.

Key definitions include Interior and exterior, Closure, Boundary. Representative theorem/result blocks include Theorem - Interior is the largest open subset, Interior laws, Kuratowski closure laws, Boundary identities, Three-way decomposition, Kuratowski closure laws. The page should list only results genuinely present in the final resource and should preserve mathematical notation rather than paraphrase theorem identities away.

A useful visible mathematical anchor is ∂ A=A∖ Acirc,   X=Acirc,cup,∂ A,cup,Ext(A).. This should render with the site's normal LaTeX/MathJax system and be accompanied by explanatory HTML text rather than embedded only in an image.

It covers Accumulation points, derived sets, isolated/perfect sets, density, nowhere density and separability.. Its visible sections are Derived Sets and Isolated Points, Perfect, Dense and Nowhere-Dense Sets, Separable Spaces and Countable Dense Sets, Derived-Set Algebra and Perfect-Set Structure. The complete PDF page range is 40–44.

Key definitions include Derived set, Perfect and dense sets, Nowhere dense, Separable space, Perfect set. Representative theorem/result blocks include Theorem - Closure decomposition, Metric sequential test for limit points, Density criterion, Theorem - \mathbb R^n is separable, Theorem - Derived set of a finite union, Theorem - In a T_1 space the derived set is closed. The page should list only results genuinely present in the final resource and should preserve mathematical notation rather than paraphrase theorem identities away.

A useful visible mathematical anchor is A' = {x: (U∖{x})∩ A≠∅ for every neighborhood Uni x}.. This should render with the site's normal LaTeX/MathJax system and be accompanied by explanatory HTML text rather than embedded only in an image.

It covers Basis criteria, subbases, generated topologies, finer/coarser comparison and standard examples.. Its visible sections are Basis Criterion and Generated Topology, Subbases and Finite-Intersection Generation, Finer and Coarser Topologies, Basis and Subbasis Generation Theorems. The complete PDF page range is 45–49.

Key definitions include Basis, Subbasis, Comparison of topologies. Representative theorem/result blocks include Theorem - A basis generates a topology, Theorem - Smallest topology containing a subbasis, Identity-map criterion, Basis criterion, Subbasis generation theorem, Theorem - Rational intervals form a countable basis for \mathbb R. The page should list only results genuinely present in the final resource and should preserve mathematical notation rather than paraphrase theorem identities away.

A useful visible mathematical anchor is τ_B={U⊆ X:(∀ x∈ U)(∃ B∈B)(x∈ B⊆ U)}.. This should render with the site's normal LaTeX/MathJax system and be accompanied by explanatory HTML text rather than embedded only in an image.

It covers Neighborhood bases, first and second countability, separability and sequential consequences.. Its visible sections are Local Bases and First Countability, Second Countability and Separability, Sequential Characterization of Closure in First-Countable Spaces, Countability Implications and Metric Equivalences. The complete PDF page range is 50–55.

Key definitions include Local base at a point, First-countable space, Second-countable space. Representative theorem/result blocks include Theorem - Every metric space is first countable, Theorem - Second countable implies first countable, Theorem - Second countable implies separable, Theorem - Closure by sequences, Theorem - Second countable implies first countable, Theorem - Second countable implies Lindelof. The page should list only results genuinely present in the final resource and should preserve mathematical notation rather than paraphrase theorem identities away.

A useful visible mathematical anchor is second countable⇒first countable,   metric: separable⇔second countable⇔Lindelof.. This should render with the site's normal LaTeX/MathJax system and be accompanied by explanatory HTML text rather than embedded only in an image.

It covers Relative topology, inherited bases, closure/interior formulas and hereditary properties.. Its visible sections are Relative Topology and Basic Open Sets, Closure and Interior in a Subspace, Hereditary Properties, Deep Mathematical Expansion: Relative Closure, Interior and Boundary. The complete PDF page range is 56–60.

Key definitions include Subspace topology. Representative theorem/result blocks include Theorem - \tau_Y is a topology, Subspace closure formula, Subspace interior warning, Theorem - Hausdorffness is hereditary, Theorem - First and second countability are hereditary, Theorem - Three equivalent tests for relative closure. The page should list only results genuinely present in the final resource and should preserve mathematical notation rather than paraphrase theorem identities away.

A useful visible mathematical anchor is τ_Y={Y∩ U:U∈τ_X},   A^,Y=Y∩A^,X.. This should render with the site's normal LaTeX/MathJax system and be accompanied by explanatory HTML text rather than embedded only in an image.

It covers Equivalent continuity criteria, open/closed maps, homeomorphisms and topological invariants.. Its visible sections are Open-Set, Closed-Set and Neighborhood Criteria, Closure Criterion and Composition, Homeomorphisms, Open Maps and Invariants, Deep Mathematical Expansion: Equivalent Continuity Criteria. The complete PDF page range is 61–65.

Key definitions include Homeomorphism. Representative theorem/result blocks include Theorem - Equivalent definitions of continuity, Theorem - Closure criterion, Criterion - Continuous bijective open map, Theorem - Closure criterion for continuity, Worked homeomorphism - (0,1) and \mathbb R. The page should list only results genuinely present in the final resource and should preserve mathematical notation rather than paraphrase theorem identities away.

A useful visible mathematical anchor is f continuous⇔ f^-1(V) open for every open V,   (0,1)congR.. This should render with the site's normal LaTeX/MathJax system and be accompanied by explanatory HTML text rather than embedded only in an image.

It covers Finite and arbitrary products, basis sets, projections and coordinatewise continuity.. Its visible sections are Basis for the Product Topology, Projections and Coordinatewise Continuity, Euclidean Products and Finite Product Metrics, Deep Mathematical Expansion: Product Metrics, Closure and Interior. The complete PDF page range is 66–70.

Key definitions include Product topology on X\times Y. Representative theorem/result blocks include Theorem - Rectangles satisfy the basis criterion, Theorem - Projection maps are continuous and open, Coordinate criterion, Theorem - Product topology on \mathbb R^m\times\mathbb R^n, Theorem - Maximum metric generates the finite product topology, Theorem - Closure of a rectangular product. The page should list only results genuinely present in the final resource and should preserve mathematical notation rather than paraphrase theorem identities away.

A useful visible mathematical anchor is B={U× V:U∈τ_X,V∈τ_Y},   p_X(x,y)=x.. This should render with the site's normal LaTeX/MathJax system and be accompanied by explanatory HTML text rather than embedded only in an image.

It covers Equivalence relations, quotient maps, universal property and identification constructions.. Its visible sections are Equivalence Relations and Quotient Topology, Universal Property of Quotient Maps, Identification of an Interval to a Circle, Deep Mathematical Expansion: Quotient Universal Property and Circle Identification. The complete PDF page range is 71–75.

Key definitions include Quotient topology, Saturated set. Representative theorem/result blocks include Theorem - Quotient topology is a topology, Universal property, Theorem - Quotient universal property in both directions, Factorization theorem through equivalence classes, Theorem - Identifying the endpoints of [0,1] gives the circle. The page should list only results genuinely present in the final resource and should preserve mathematical notation rather than paraphrase theorem identities away.

A useful visible mathematical anchor is U⊆ X/∼ open⇔ q^-1(U) open in X,   [0,1]/(0∼1)cong S¹.. This should render with the site's normal LaTeX/MathJax system and be accompanied by explanatory HTML text rather than embedded only in an image.

It covers Dense countable sets, Lindelof spaces, second countability and metric equivalences.. Its visible sections are Lindelof Spaces and Countable Subcovers, Metric Equivalence: Separable, Second Countable, Lindelof, Lindelof Metric Implies Separable, Deep Mathematical Expansion: Countability Equivalences in Metric Spaces. The complete PDF page range is 76–80.

Key definitions include Lindelof space. Representative theorem/result blocks include Theorem - Second countable implies Lindelof, Theorem - Separable metric implies second countable, Theorem - Lindelof metric spaces are separable, Theorem - Second countable implies Lindelof, Theorem - Separable metric implies second countable, Theorem - Lindelof metric implies separable. The page should list only results genuinely present in the final resource and should preserve mathematical notation rather than paraphrase theorem identities away.

A useful visible mathematical anchor is second countable⇒Lindelof, second countable⇒separable.. This should render with the site's normal LaTeX/MathJax system and be accompanied by explanatory HTML text rather than embedded only in an image.

It covers Sequence convergence, cluster points, first-countable criteria and limits beyond metric spaces.. Its visible sections are Convergence of Sequences in Topological Spaces, Sequential Continuity and First-Countable Spaces, Why Sequences Are Not Enough, Deep Mathematical Expansion: Sequential Tests in First-Countable Spaces. The complete PDF page range is 81–85.

Key definitions include Topological convergence. Representative theorem/result blocks include Examples of non-metric behavior, Theorem - Continuity implies sequential continuity, Theorem - In first-countable domains, sequential continuity implies continuity, Counterexample - Cocountable topology, Theorem - Closure is sequential in first-countable spaces, Theorem - Sequential continuity is continuity in first-countable domains. The page should list only results genuinely present in the final resource and should preserve mathematical notation rather than paraphrase theorem identities away.

A useful visible mathematical anchor is x_n→ x⇔ (∀ U∈N(x))(∃ N)(n≥ N⇒ x_n∈ U).. This should render with the site's normal LaTeX/MathJax system and be accompanied by explanatory HTML text rather than embedded only in an image.

It covers Kolmogorov, Frechet and Hausdorff conditions, singleton tests, diagonals and uniqueness of limits.. Its visible sections are The T0 and T1 Axioms, Hausdorff Spaces and Uniqueness of Limits, Diagonal Characterization and Preservation, Deep Mathematical Expansion: Singleton, Diagonal and Preservation Criteria. The complete PDF page range is 86–90.

Key definitions include T_0 and T_1, Hausdorff space. Representative theorem/result blocks include Theorem - T_1 iff singletons are closed, Theorem - Sequence limits are unique in Hausdorff spaces, Theorem - Hausdorff iff the diagonal is closed, Theorem - T_1 iff every singleton is closed, Theorem - Hausdorff iff the diagonal is closed, Corollary - Compact subsets of Hausdorff spaces are closed. The page should list only results genuinely present in the final resource and should preserve mathematical notation rather than paraphrase theorem identities away.

A useful visible mathematical anchor is T₂:;x≠ y⇒∃ Uni x,Vni y,;U∩ V=∅,   Δ_X closed⇔ X Hausdorff.. This should render with the site's normal LaTeX/MathJax system and be accompanied by explanatory HTML text rather than embedded only in an image.

It covers T3, Tychonoff and T4 separation, shrinking criteria and metric-space separation.. Its visible sections are Regular Spaces and Shrinking, Metric Spaces Are Normal, Complete Regularity and Separation Hierarchy, Deep Mathematical Expansion: Metric Separation by Distance Functions. The complete PDF page range is 91–95.

Key definitions include Regular space, Completely regular space. Representative theorem/result blocks include Shrinking criterion, Theorem - Every metric space is normal, Theorem - Metric spaces are completely regular, Lemma - Distance to a closed set is positive off the set locally, Theorem - Every metric space is normal, Theorem - Metric spaces are completely regular. The page should list only results genuinely present in the final resource and should preserve mathematical notation rather than paraphrase theorem identities away.

A useful visible mathematical anchor is T₄⇒ T₃₁₂⇒ T₃⇒ T₂⇒ T₁⇒ T₀.. This should render with the site's normal LaTeX/MathJax system and be accompanied by explanatory HTML text rather than embedded only in an image.

It covers Urysohn functions, dyadic constructions, embedding ideas and Urysohn metrization.. Its visible sections are Urysohn Lemma: Dyadic Construction, Regular Second-Countable Spaces Are Normal, Urysohn Metrization Theorem, Deep Mathematical Expansion: Urysohn Function and Explicit Metrization. The complete PDF page range is 96–100.

Key definitions include the chapter-specific objects introduced in the section text. Representative theorem/result blocks include Urysohn Lemma, Theorem - Regular Lindelof implies normal, Urysohn Metrization Theorem, Urysohn construction - nested dyadic open sets, Theorem - Urysohn function from the dyadic family, Explicit metric from a countable separating family. The page should list only results genuinely present in the final resource and should preserve mathematical notation rather than paraphrase theorem identities away.

A useful visible mathematical anchor is ρ(x,y)=∑_n=1^∞2^-n|f_n(x)-f_n(y)|.. This should render with the site's normal LaTeX/MathJax system and be accompanied by explanatory HTML text rather than embedded only in an image.

It covers Open covers, finite subcovers, FIP, Hausdorff consequences and continuous images.. Its visible sections are Open Covers and Compact Spaces, Finite Intersection Property, Compactness and Hausdorff Spaces, Deep Mathematical Expansion: FIP, Closedness and Compact-Hausdorff Bijections. The complete PDF page range is 101–105.

Key definitions include Compact space, Finite intersection property. Representative theorem/result blocks include Theorem - Closed subsets of compact spaces are compact, FIP characterization of compactness, Theorem - Compact subsets of Hausdorff spaces are closed, Corollary - Compact-to-Hausdorff bijection, Theorem - Compactness and the finite-intersection property, Theorem - Compact subsets of Hausdorff spaces are closed. The page should list only results genuinely present in the final resource and should preserve mathematical notation rather than paraphrase theorem identities away.

A useful visible mathematical anchor is X compact⇔every open cover has a finite subcover⇔FIP characterization.. This should render with the site's normal LaTeX/MathJax system and be accompanied by explanatory HTML text rather than embedded only in an image.

It covers Sequential compactness, limit-point compactness, countable compactness, Lebesgue numbers, total boundedness and completeness.. Its visible sections are Sequential Compactness and Limit-Point Compactness, Lebesgue Numbers Without Circular Reasoning, \varepsilon-Nets, Total Boundedness and Cauchy Subsequences, Complete Metric Compactness Equivalence. The complete PDF page range is 106–113.

Key definitions include Sequentially compact metric space, Limit-point compactness, Lebesgue number of an open cover, \varepsilon-net and totally bounded metric space, Countably compact space. Representative theorem/result blocks include Lemma - A metric limit point sees infinitely many points, Theorem - Sequential compactness \Longleftrightarrow limit-point compactness in metric spaces, Lebesgue Number Lemma - Sequentially compact form, Equivalent ball formulation, Theorem - Sequential compactness implies total boundedness, Theorem - Complete plus totally bounded implies sequentially compact. The page should list only results genuinely present in the final resource and should preserve mathematical notation rather than paraphrase theorem identities away.

A useful visible mathematical anchor is metric compact⇔sequentially compact⇔complete + totally bounded.. This should render with the site's normal LaTeX/MathJax system and be accompanied by explanatory HTML text rather than embedded only in an image.

It covers Relatively compact neighborhoods, regularity, compact Hausdorff normality, Alexandroff compactification and convergence to infinity.. Its visible sections are Local Compactness and Relatively Compact Neighborhood Bases, Compact Hausdorff Spaces Are Normal, Construction of the One-Point Compactification, Uniqueness, Convergence to Infinity and the Circle Model. The complete PDF page range is 114–120.

Key definitions include Local compactness, Alexandroff one-point compactification. Representative theorem/result blocks include Theorem - Locally compact Hausdorff spaces are regular, Corollary - Relatively compact open sets form a local base, Hereditary results, Lemma - A point and a compact set can be separated, Theorem - Compact Hausdorff implies normal, Corollary - Compact Hausdorff spaces are completely regular. The page should list only results genuinely present in the final resource and should preserve mathematical notation rather than paraphrase theorem identities away.

A useful visible mathematical anchor is X^*=X∪{∞},   N(∞)={X^*∖ K:K⊆ X compact}.. This should render with the site's normal LaTeX/MathJax system and be accompanied by explanatory HTML text rather than embedded only in an image.

It covers Separations, clopen criteria, connected images, intervals, components, products and total disconnectedness.. Its visible sections are Separations, Clopen Sets and Union Theorems, Connected Subsets of the Real Line, Components, Partitions and Total Disconnectedness, Products and Further Preservation Theorems. The complete PDF page range is 121–127.

Key definitions include Separation and connectedness, Connected component, Totally disconnected space. Representative theorem/result blocks include Clopen criterion, Theorem - Union of connected sets with a common point, Theorem - Closure preserves connectedness, Theorem - Every interval in \mathbb R is connected, Theorem - Connected subsets of \mathbb R are exactly intervals, Theorem - Continuous images preserve connectedness. The page should list only results genuinely present in the final resource and should preserve mathematical notation rather than paraphrase theorem identities away.

A useful visible mathematical anchor is X connected⇔ X has no nontrivial clopen subset.. This should render with the site's normal LaTeX/MathJax system and be accompanied by explanatory HTML text rather than embedded only in an image.

It covers Paths, path operations, path components, convexity, local connectedness and the topologist's sine curve.. Its visible sections are Paths, Reversal and Concatenation, Convexity, Products and Common-Point Unions, Local Connectedness and Local Path Connectedness, The Topologist's Sine Curve - Connected but Not Path Connected. The complete PDF page range is 128–135.

Key definitions include Path and path-connected space, Convex subset of \mathbb R^n, Local connectedness and local path connectedness, Topologist's sine curve. Representative theorem/result blocks include Theorem - Path connected implies connected, Path reversal, Path concatenation, Path relation is an equivalence relation, Theorem - Convex sets are path connected, Theorem - Products preserve path connectedness. The page should list only results genuinely present in the final resource and should preserve mathematical notation rather than paraphrase theorem identities away.

A useful visible mathematical anchor is γ:[0,1]→ X, γ(0)=x,;γ(1)=y,   path connected⇒connected.. This should render with the site's normal LaTeX/MathJax system and be accompanied by explanatory HTML text rather than embedded only in an image.

It covers Nowhere dense sets, meagre and residual sets, Baire spaces, equivalent formulations and the complete-metric Baire Category Theorem.. Its visible sections are Nowhere Dense, Meagre and Residual Sets, Baire Spaces and Equivalent Formulations, Baire Category Theorem for Complete Metric Spaces, Concrete Consequences of Baire Category. The complete PDF page range is 136–142.

Key definitions include Nowhere dense set, Meagre, first category, residual and second category, Baire space. Representative theorem/result blocks include Equivalent nowhere-dense criterion, Elementary category algebra, Master equivalence for Baire spaces, Theorem - Open subspaces of Baire spaces are Baire, Baire Category Theorem, Corollary - \mathbb R is not countable. The page should list only results genuinely present in the final resource and should preserve mathematical notation rather than paraphrase theorem identities away.

A useful visible mathematical anchor is X Baire⇔bigcap_n=1^∞U_n is dense whenever every U_n is open dense.. This should render with the site's normal LaTeX/MathJax system and be accompanied by explanatory HTML text rather than embedded only in an image.

It covers Directed sets, nets, subnets, cluster points and net characterizations of closure, continuity and compactness.. Its visible sections are Directed Sets and Nets, Closure and Continuity via Nets, Subnets, Cluster Points and Compactness. The complete PDF page range is 142–148.

Key definitions include Directed set and net, Convergence of a net, Eventually and frequently, Subnet, Cluster point of a net. Representative theorem/result blocks include Proposition - Products of directed sets are directed, Proposition - Neighborhoods form a directed set, Theorem - Net characterization of closure, Theorem - Continuity via nets, Theorem - Closed sets are exactly net-closed sets, Theorem - Continuity at a point via nets. The page should list only results genuinely present in the final resource and should preserve mathematical notation rather than paraphrase theorem identities away.

A useful visible mathematical anchor is xalpha→ x⇔ (∀ U∈N(x))(∃α₀)(αsucceqα₀⇒ xalpha∈ U).. This should render with the site's normal LaTeX/MathJax system and be accompanied by explanatory HTML text rather than embedded only in an image.

It covers Filters, filter bases, convergence, adherence, ultrafilters and compactness characterizations.. Its visible sections are Filters and Filter Bases, Filter Convergence and Net-Filter Correspondence, Ultrafilters and Compactness. The complete PDF page range is 149–156.

Key definitions include Filter, Filter base, Principal and cofinite filters, Neighborhood filter, Filter convergence, Adherence point of a filter. Representative theorem/result blocks include Theorem - A filter base generates a filter, Theorem - Tail filter of a net, Proposition - Filter convergence implies adherence, Theorem - Continuity carries convergent filters to convergent filters, Theorem - A canonical net associated with a filter, Ultrafilter dichotomy. The page should list only results genuinely present in the final resource and should preserve mathematical notation rather than paraphrase theorem identities away.

A useful visible mathematical anchor is F ultrafilter⇔(∀ A⊆ X)(A∈F or X∖ A∈F).. This should render with the site's normal LaTeX/MathJax system and be accompanied by explanatory HTML text rather than embedded only in an image.

It covers Arbitrary products, ultrafilter proof of Tychonoff and the Alexander subbase method.. Its visible sections are Arbitrary Product Topology, Tychonoff Theorem via Ultrafilters, Alexander Subbase Theorem. The complete PDF page range is 157–164.

Key definitions include Product topology. Representative theorem/result blocks include Coordinatewise convergence of nets, Theorem - Finite-support rectangles form a basis, Universal property of the product topology, Proposition - Hausdorff products, Metric for a countable product of metric spaces, Tychonoff Theorem. The page should list only results genuinely present in the final resource and should preserve mathematical notation rather than paraphrase theorem identities away.

A useful visible mathematical anchor is ∏_{i∈ I}X_i compact if every X_i is compact (Tychonoff).. This should render with the site's normal LaTeX/MathJax system and be accompanied by explanatory HTML text rather than embedded only in an image.

It covers Compactifications, one-point comparison, Tychonoff embedding and Stone-Cech construction viewpoint.. Its visible sections are Compactifications and Equivalence, Tychonoff Embedding Theorem, Stone-Cech Construction Viewpoint. The complete PDF page range is 165–170.

Key definitions include Compactification, Equivalent compactifications. Representative theorem/result blocks include Dense-agreement principle, Proposition - Equivalence of compactifications is an equivalence relation, Proposition - Equivalent compactifications have homeomorphic remainders, Tychonoff Embedding Theorem, Point and closed-set separation in a Tychonoff space, Corollary - Existence of compactifications. The page should list only results genuinely present in the final resource and should preserve mathematical notation rather than paraphrase theorem identities away.

A useful visible mathematical anchor is e:X→[0,1]^C(X,[0,1]),   e(x)(f)=f(x),   β X=e(X).. This should render with the site's normal LaTeX/MathJax system and be accompanied by explanatory HTML text rather than embedded only in an image.

It covers Locally finite refinements, paracompactness, normality consequences and metrization criteria.. Its visible sections are Locally Finite Families and Refinements, Paracompactness and Normality, Sigma-Locally Finite Bases and Metrization. The complete PDF page range is 171–177.

Key definitions include Locally finite family, Refinement, Paracompact space, \sigma-locally finite basis. Representative theorem/result blocks include Theorem - Locally finite unions of closed sets are closed, Proposition - Subfamilies of locally finite families are locally finite, Proposition - Locally finite closure preservation, Corollary - Closure of a locally finite union, Theorem - Regular Lindelof spaces are paracompact, Proposition - Compact spaces are paracompact. The page should list only results genuinely present in the final resource and should preserve mathematical notation rather than paraphrase theorem identities away.

A useful visible mathematical anchor is paracompact:;∀U;∃ locally finite open refinement V.. This should render with the site's normal LaTeX/MathJax system and be accompanied by explanatory HTML text rather than embedded only in an image.

It covers Sigma-locally finite refinements and the detailed proof that every metric space is paracompact.. Its visible sections are Distance Layers and Sigma-Locally Finite Refinements, Stone's Theorem - Every Metric Space Is Paracompact, Consequences and Preservation Results. The complete PDF page range is 178–184.

Key definitions include Distance to the complement and metric depth. Representative theorem/result blocks include Lemma - The distance-to-complement function is 1-Lipschitz, Lemma - Positivity detects membership in an open set, Lemma - A quantitative gap between consecutive layers, Stone's Theorem, Corollary - Metric spaces are paracompact Hausdorff and normal, Theorem - Closed subspaces of paracompact spaces are paracompact. The page should list only results genuinely present in the final resource and should preserve mathematical notation rather than paraphrase theorem identities away.

A useful visible mathematical anchor is δ_U(x)=d(x,X∖ U),   |δ_U(x)-δ_U(y)|≤ d(x,y).. This should render with the site's normal LaTeX/MathJax system and be accompanied by explanatory HTML text rather than embedded only in an image.

It covers Normality, shrinkings, locally finite sums, supports and subordinate partitions of unity.. Its visible sections are Paracompact Hausdorff Implies Regular and Normal, Shrinkings and Locally Finite Sums, Partitions of Unity. The complete PDF page range is 185–191.

Key definitions include Shrinking of an open cover, Support and partition of unity. Representative theorem/result blocks include Lemma - Closure commutes with a locally finite union, Theorem - Every paracompact Hausdorff space is regular, Theorem - Paracompact Hausdorff implies normal, Corollary - Compact Hausdorff spaces are normal, Shrinking theorem for paracompact Hausdorff spaces, Two-stage shrinking. The page should list only results genuinely present in the final resource and should preserve mathematical notation rather than paraphrase theorem identities away.

A useful visible mathematical anchor is ϕ_i≥0,   ∑_iϕ_i(x)=1,   suppϕ_i⊆ U_i.. This should render with the site's normal LaTeX/MathJax system and be accompanied by explanatory HTML text rather than embedded only in an image.

It covers Stars of covers, star refinements, point-finite families and numerable covers.. Its visible sections are Stars of Covers, Star Refinements and Full Normality, Point-Finite and Numerable Covers. The complete PDF page range is 192–196.

Key definitions include Star of a set with respect to a family, Star refinement, Fully normal space, Point-finite and locally finite families, Numerable cover. Representative theorem/result blocks include Basic star identities, Why ordinary refinement is weaker than star refinement, Theorem - Every paracompact Hausdorff space is fully normal, Corollary - Paracompact Hausdorff \Rightarrow fully normal \Rightarrow normal, Theorem - Local finiteness implies point-finiteness, Counterexample - Point-finite need not be locally finite. The page should list only results genuinely present in the final resource and should preserve mathematical notation rather than paraphrase theorem identities away.

A useful visible mathematical anchor is St(A,U)=bigcup{U∈U:U∩ A≠∅}.. This should render with the site's normal LaTeX/MathJax system and be accompanied by explanatory HTML text rather than embedded only in an image.

It covers Pointwise and compact-open topologies, evaluation maps and uniform topology on compact domains.. Its visible sections are Pointwise and Compact-Open Topologies, Evaluation Maps and Local Compactness, Compact-Open Equals Uniform Topology on Compact Domains, Convergence and a Spike-Function Counterexample. The complete PDF page range is 197–203.

Key definitions include Pointwise topology, Compact-open topology, Uniform metric on a compact domain. Representative theorem/result blocks include Theorem - Pointwise topology is generated by evaluation maps, Theorem - Compact-open is finer than pointwise, Corollary - Equality for finite domains, Local compactness lemma, Theorem - Joint evaluation is continuous, Corollary - Every fixed-point evaluation is continuous. The page should list only results genuinely present in the final resource and should preserve mathematical notation rather than paraphrase theorem identities away.

A useful visible mathematical anchor is [K,U]={f∈ C(X,Y):f(K)⊆ U},   dinfty(f,g)=sup_{x∈ X}d(f(x),g(x)).. This should render with the site's normal LaTeX/MathJax system and be accompanied by explanatory HTML text rather than embedded only in an image.

It covers Covering dimension, order of covers, dimension zero and one, and inductive dimensions.. Its visible sections are Covering Dimension and Order of Covers, Dimension Zero, the Cantor Set and the Interval, Inductive Dimensions. The complete PDF page range is 204–209.

Key definitions include Multiplicity and order of a cover, Lebesgue covering dimension, Base convention, Small inductive dimension, Large inductive dimension. Representative theorem/result blocks include Equivalent intersection formulation, Theorem - Covering dimension is a topological invariant, Theorem - Discrete spaces have covering dimension zero, Theorem - The Cantor set has covering dimension zero, Theorem - \dim[0,1]=1, Theorem - Discrete spaces have inductive dimension zero. The page should list only results genuinely present in the final resource and should preserve mathematical notation rather than paraphrase theorem identities away.

A useful visible mathematical anchor is dim X≤ n⇔every finite open cover has a refinement of order ≤ n+1.. This should render with the site's normal LaTeX/MathJax system and be accompanied by explanatory HTML text rather than embedded only in an image.

It covers Classical spaces separating topological properties and a complete implication/counterexample map.. Its visible sections are Cofinite, Cocountable and Indiscrete Pathologies, Sorgenfrey, Sierpinski and Uncountable Product Counterexamples, Implication Map and Explicit Failure of Converses, Why Sequences Do Not Detect All Topological Closure. The complete PDF page range is 210–215.

Key definitions include the chapter-specific objects introduced in the section text. Representative theorem/result blocks include Cofinite topology - a four-property calculation, Cocountable topology on an uncountable set, Sequence theorem in the cocountable topology, Indiscrete pathology, The Sorgenfrey line, Sierpinski space - T_0 need not imply T_1. The page should list only results genuinely present in the final resource and should preserve mathematical notation rather than paraphrase theorem identities away.

A useful visible mathematical anchor is compactnot⇒Hausdorff, T₁not⇒ T₂, metricnot⇒separable.. This should render with the site's normal LaTeX/MathJax system and be accompanied by explanatory HTML text rather than embedded only in an image.

It covers Linear orders, interval bases, ordered topological spaces, compact intervals, linear continua and ordinal examples.. Its visible sections are Linear Orders, Intervals and Order-Convex Sets, The Order Topology and LOTS, Compact Intervals, Linear Continua and Ordinal Examples, Order Completeness, Compactness and Interval Connectedness. The complete PDF page range is 216–225.

Key definitions include Linear order, Order-convex subset, Order topology, Linearly ordered topological space (LOTS), Least-upper-bound property, Linear continuum. Representative theorem/result blocks include Lemma - Intersections of order-convex sets, Theorem - The order intervals form a basis, Theorem - Every LOTS is Hausdorff, Theorem - Order topology on \mathbb R, Theorem - Compactness of closed order intervals, Theorem - Linear continua are connected. The page should list only results genuinely present in the final resource and should preserve mathematical notation rather than paraphrase theorem identities away.

A useful visible mathematical anchor is (a,b)={x:a<x<b},   order topology generated by open intervals and rays.. This should render with the site's normal LaTeX/MathJax system and be accompanied by explanatory HTML text rather than embedded only in an image.

It covers Coproduct topology, universal mapping property, preservation theorems and interactions with quotient constructions.. Its visible sections are The Topological Sum, Preservation and Reflection of Topological Properties, Sums, Quotients and Wedge Constructions, Connectedness, Local Properties and Countability in Sums. The complete PDF page range is 226–232.

Key definitions include Topological sum, Wedge sum at chosen base points. Representative theorem/result blocks include Theorem - Universal property of the sum, Theorem - Hausdorffness, regularity and normality of sums, Theorem - Compactness criterion, Theorem - Second countability, Theorem - Arbitrary sums preserve paracompactness, Theorem - Continuity criterion for maps out of a wedge. The page should list only results genuinely present in the final resource and should preserve mathematical notation rather than paraphrase theorem identities away.

A useful visible mathematical anchor is bigsqcup_{i∈ I}X_i,   U open⇔ U∩ X_i open in X_i for every i.. This should render with the site's normal LaTeX/MathJax system and be accompanied by explanatory HTML text rather than embedded only in an image.

It covers Urysohn approximation, complete bounded Tietze proof, extension corollaries and normal-space function separation.. Its visible sections are The Urysohn Approximation Lemma, The Bounded Tietze Extension Theorem, Full Tietze Theorem and Function-Theoretic Consequences, Equivalence of Normality, Urysohn Separation and Tietze Extension. The complete PDF page range is 233–241.

Key definitions include the chapter-specific objects introduced in the section text. Representative theorem/result blocks include Approximation Lemma, Tietze Extension Theorem - Bounded form, Detailed induction invariant and tail estimate, Scaled bounded form, Tietze Extension Theorem - Real-valued form, Corollary - Extension to an arbitrary finite interval. The page should list only results genuinely present in the final resource and should preserve mathematical notation rather than paraphrase theorem identities away.

A useful visible mathematical anchor is f:A→[-1,1] continuous,;A closed in normal X⇒∃ F:X→[-1,1],;F|_A=f.. This should render with the site's normal LaTeX/MathJax system and be accompanied by explanatory HTML text rather than embedded only in an image.

It covers Entourages, metric uniformities, uniform continuity, products, Cauchy filters, completeness and completion.. Its visible sections are Uniformities and the Topology Generated by Entourages, Uniform Continuity and Product Uniformities, Cauchy Filters, Cauchy Nets and Completeness, Total Boundedness and Compactness, Uniform Equivalence, Completeness Invariance and Completion Mapping Property. The complete PDF page range is 242–255.

Key definitions include Uniformity, Uniformly continuous map, Product uniformity, Cauchy filter, Cauchy net, Complete uniform space. Representative theorem/result blocks include Theorem - Every uniformity induces a topology, Theorem - Metric uniformity, Theorem - Uniform continuity implies continuity, Metric translation, Theorem - The product uniformity induces the product topology, Theorem - Convergent filters are Cauchy. The page should list only results genuinely present in the final resource and should preserve mathematical notation rather than paraphrase theorem identities away.

A useful visible mathematical anchor is U[x]={y:(x,y)∈ U},   V∘ V⊆ U,   X=C/∼.. This should render with the site's normal LaTeX/MathJax system and be accompanied by explanatory HTML text rather than embedded only in an image.

It covers Local metrizability, collectionwise normality, Smirnov, Bing and Nagata-Smirnov viewpoints.. Its visible sections are Locally Metrizable Spaces, Collectionwise Normality and Discrete Families, A Metrization Implication Network, Paracompact Local Metrizability and a Global Metrization Proof. The complete PDF page range is 256–262.

Key definitions include Locally metrizable space, Discrete family, Collectionwise normal space. Representative theorem/result blocks include Theorem - Topological sums of metrizable spaces are metrizable, Smirnov Metrization Theorem, Theorem - Paracompact Hausdorff spaces are collectionwise normal, Corollary - Metric spaces are collectionwise normal, Bing-Nagata-Smirnov perspective, Theorem - Paracompact Hausdorff locally metrizable spaces are metrizable. The page should list only results genuinely present in the final resource and should preserve mathematical notation rather than paraphrase theorem identities away.

A useful visible mathematical anchor is paracompact Hausdorff + locally metrizable⇒metrizable.. This should render with the site's normal LaTeX/MathJax system and be accompanied by explanatory HTML text rather than embedded only in an image.

It covers Charts, atlases, manifolds with boundary, metrizability, connectedness, compact examples, connected sums and elementary surgery ideas.. Its visible sections are Topological Manifolds, Charts and Atlases, Basic Examples and Global Consequences, Manifolds with Boundary, Connected Sum and Elementary Surgery, Countable Atlases, Product Manifolds and Compactness Consequences. The complete PDF page range is 263–271.

Key definitions include n-dimensional topological manifold, Chart, Manifold with boundary, Connected sum, Elementary topological surgery idea. Representative theorem/result blocks include Theorem - Open subspaces of \mathbb R^n are n-manifolds, Theorem - Every topological manifold is locally compact and locally path connected, Theorem - The sphere S^n is an n-manifold, Theorem - Every topological manifold is metrizable, Theorem - Connected manifolds are path connected, Brouwer Invariance of Dimension - Statement. The page should list only results genuinely present in the final resource and should preserve mathematical notation rather than paraphrase theorem identities away.

A useful visible mathematical anchor is ϕ:U→ϕ(U)⊆Rⁿ,   ϕbeta∘ϕalpha^-1 on overlaps.. This should render with the site's normal LaTeX/MathJax system and be accompanied by explanatory HTML text rather than embedded only in an image.

It covers Universal constructions generated by families of maps; products, subspaces, sums and quotients as special cases.. Its visible sections are Initial Topologies and Weak Topologies, Products and Subspaces as Initial Constructions, Final Topologies, Quotients and Sums, Embedding Criteria and Calculus of Initial/Final Constructions. The complete PDF page range is 272–278.

Key definitions include Initial topology, Weak topology generated by a family of functions, Final topology. Representative theorem/result blocks include Theorem - Subbasis for the initial topology, Universal Property - Initial topology, Theorem - Product topology is initial, Universal Property - Product topology, Theorem - Subspace topology is initial, Theorem - Existence and maximality of the final topology. The page should list only results genuinely present in the final resource and should preserve mathematical notation rather than paraphrase theorem identities away.

A useful visible mathematical anchor is τ_init=σ({f_i^-1(U):U open}),  τ_final={V:g_j^-1(V) open ∀ j}.. This should render with the site's normal LaTeX/MathJax system and be accompanied by explanatory HTML text rather than embedded only in an image.

It covers Lower-limit, upper-limit and K-topologies; detailed comparison, convergence and separation counterexamples.. Its visible sections are The Lower-Limit and Upper-Limit Topologies, The K-Topology, Comparison Maps and Convergence Diagnostics, Zero-Dimensional Features and Sharper Convergence Tests. The complete PDF page range is 279–288.

Key definitions include Lower-limit topology, Upper-limit topology, K-topology. Representative theorem/result blocks include Theorem - The lower-limit topology is strictly finer than the usual topology, Theorem - First countability of the Sorgenfrey line, Theorem - Separability but failure of second countability, Theorem - The Sorgenfrey line is Hausdorff, Corollary - The Sorgenfrey line is not metrizable, Theorem - Sequence convergence in the Sorgenfrey line. The page should list only results genuinely present in the final resource and should preserve mathematical notation rather than paraphrase theorem identities away.

A useful visible mathematical anchor is Rell:;[a,b) basis;   R_K:;(a,b) and (a,b)∖ K basic neighborhoods.. This should render with the site's normal LaTeX/MathJax system and be accompanied by explanatory HTML text rather than embedded only in an image.

It covers Interval fixed points, Darboux-type consequences, no-retraction arguments and compact connected subsets of the real line.. Its visible sections are The Interval Fixed-Point Theorem, Connected Images, Darboux Property and No-Retraction Arguments, Compact Connected Subsets of the Real Line, Connected Targets, Locally Constant Maps and a General Sign-Change Principle. The complete PDF page range is 289–295.

Key definitions include Retraction. Representative theorem/result blocks include Fixed-Point Theorem for a Closed Interval, Theorem - Continuous images of intervals are intervals, Darboux-type consequence, Theorem - No retraction of an interval onto its two endpoints, Theorem - Classification of nonempty compact connected subsets of \mathbb R, Corollary. The page should list only results genuinely present in the final resource and should preserve mathematical notation rather than paraphrase theorem identities away.

A useful visible mathematical anchor is g(x)=f(x)-x, g(a)≥0,;g(b)≤0⇒∃ c∈[a,b]:g(c)=0.. This should render with the site's normal LaTeX/MathJax system and be accompanied by explanatory HTML text rather than embedded only in an image.