Munkres, Topology

SRC-TEXT-MUN00

Area: Topology
Status: draft
Source

Contents

586 problems.

1

Distributive laws and DeMorgan’s laws for sets — Munkres §1.1
Subset membership under union and intersection — Munkres §1.2
Contrapositive, converse, and truth of conditional statements — Munkres §1.3
Negation of quantified statements about real numbers — Munkres §1.4
Membership in unions and intersections of collections — Munkres §1.5
Contrapositives of quantified subset statements — Munkres §1.6
Expressing sets using union, intersection, and difference — Munkres §1.7
Power set of a finite set — Munkres §1.8
DeMorgan’s laws for arbitrary unions and intersections — Munkres §1.9
Cartesian products as subsets of \(\RR \times \RR\) — Munkres §1.10

2

Preimage and image under injective and surjective maps — Munkres §2.1
Preimage preserves all set operations; image preserves unions only — Munkres §2.2
Preimage and image for arbitrary unions and intersections — Munkres §2.3
Composition of functions and injectivity and surjectivity — Munkres §2.4
Left inverses, right inverses, and bijectivity — Munkres §2.5
Restricting domain and range to obtain a bijection — Munkres §2.6

4

Laws of algebra for \(\mathbb{R}\) from the field axioms — Munkres §4.1
Laws of inequalities for \(\mathbb{R}\) — Munkres §4.2
Inductive sets and basic properties of \(\mathbb{Z}_+\) — Munkres §4.3
Largest element of finite subsets of \(\{1, \ldots, n\}\) — Munkres §4.4
Closure properties of \(\mathbb{Z}\) and \(\mathbb{Z}_+\) — Munkres §4.5
Laws of exponents for real numbers — Munkres §4.6
Extending the laws of exponents to all integers — Munkres §4.7
Greatest lower bound property of \(\mathbb{R}\) — Munkres §4.8
Archimedean property and density of \(\mathbb{Q}\) in \(\mathbb{R}\) — Munkres §4.9
Existence and uniqueness of positive square roots — Munkres §4.10
Even and odd integers and irrationality of \(\sqrt{2}\) — Munkres §4.11

5

Bijection between \(A \times B\) and \(B \times A\) — Munkres §5.1
Associativity of finite and infinite Cartesian products — Munkres §5.2
Containment and nonemptiness of infinite products — Munkres §5.3
Bijections between finite and countable product spaces — Munkres §5.4
Subsets of \(\mathbb{R}^{\omega}\) as Cartesian products — Munkres §5.5

6

Injective maps between finite sets of different sizes — Munkres §6.1
Subsets of finite sets are finite — Munkres §6.2
Cantor’s theorem for \(X^{\omega}\) — Munkres §6.3
Finite simply ordered sets have a largest element — Munkres §6.4
Finiteness of Cartesian products — Munkres §6.5
Bijection between \(\mathcal{P}(A)\) and \(X^n\) — Munkres §6.6
Finiteness of function sets between finite sets — Munkres §6.7

7

Countability of \(\mathbb{Q}\) — Munkres §7.1
Bijections in the countability proofs for \(\mathbb{Q}\) — Munkres §7.2
Bijection between \(\mathcal{P}(\mathbb{Z}_+)\) and \(X^{\omega}\) — Munkres §7.3
Countability of algebraic numbers and uncountability of transcendentals — Munkres §7.4
Countability of function spaces and subsets of \(\mathbb{Z}_+\) — Munkres §7.5
Schroeder–Bernstein theorem and equal cardinality — Munkres §7.6
Equal cardinality of \(\{0,1\}^{\omega}\) subsets — Munkres §7.7
Cardinality of countable subsets of \(X^{\omega}\) — Munkres §7.8
Recursion formulas that fall outside the standard principle — Munkres §7.9

9

Injective map from \(\mathbb{Z}_+\) to \(X^{\omega}\) without choice — Munkres §9.1
Choice functions for countable collections — Munkres §9.2
Infinite sets from sequences of injections — Munkres §9.3
Identifying implicit use of the axiom of choice — Munkres §9.4
Right inverses from surjections via the axiom of choice — Munkres §9.5
Paradoxes of the set of all sets — Munkres §9.6
Strict cardinality ordering and the continuum — Munkres §9.7

10

Well-ordered sets have the least upper bound property — Munkres §10.1
Immediate successors in well-ordered sets — Munkres §10.2
Order types of \(\{1,2\} \times \ZZ_+\) and \(\ZZ_+ \times \{1,2\}\) — Munkres §10.3
Characterizing non-well-ordered sets via \(\ZZ_{-}\) — Munkres §10.4
Well-ordering theorem implies the axiom of choice — Munkres §10.5
Properties of the minimal uncountable well-ordered set \(S_{\Omega}\) — Munkres §10.6
Principle of transfinite induction — Munkres §10.7
Well-ordering unions of disjoint well-ordered sets — Munkres §10.8
Antidictionary order on sequences ending in \(1\)’s — Munkres §10.9
Transfinite recursion on well-ordered sets — Munkres §10.10
Comparing cardinalities using the well-ordering theorem — Munkres §10.11

13

Sets containing a neighborhood of each of their points are open — Munkres §13.1
Comparing the nine topologies on a three-point set — Munkres §13.2
The countable complement topology and an infinite complement collection — Munkres §13.3
Intersections and unions of families of topologies — Munkres §13.4
Generated topologies as intersections of topologies — Munkres §13.5
The lower limit and K-topologies are not comparable — Munkres §13.6
Comparing five topologies on the real line — Munkres §13.7
Countable rational-endpoint bases — Munkres §13.8

16

Transitivity of the subspace topology — Munkres §16.1
Refining a topology refines the subspace topologies — Munkres §16.2
Open sets in a closed interval versus open sets in the line — Munkres §16.3
Projections from a product are open maps — Munkres §16.4
Product topologies under refinement of the factors — Munkres §16.5
A countable rational rectangle basis for the plane — Munkres §16.6
Convex subsets that are not intervals or rays — Munkres §16.7
Subspace topologies on lines in the Sorgenfrey plane — Munkres §16.8
Dictionary order topology on the plane as a product topology — Munkres §16.9
Three topologies on the unit square — Munkres §16.10

17

Closed sets satisfy the topology axioms — Munkres §17.1
Closed subsets of closed subspaces are closed — Munkres §17.2
Products of closed sets are closed — Munkres §17.3
Differences of open and closed sets — Munkres §17.4
Closure of an interval in the order topology — Munkres §17.5
Closure behavior under unions — Munkres §17.6
Where a proof about closures of unions fails — Munkres §17.7
Closures of intersections and differences — Munkres §17.8
Closure of a product equals the product of closures — Munkres §17.9
Order topologies are Hausdorff — Munkres §17.10
Products of Hausdorff spaces are Hausdorff — Munkres §17.11
Subspaces of Hausdorff spaces are Hausdorff — Munkres §17.12
Hausdorff spaces and the closed diagonal — Munkres §17.13
Convergence of a sequence in the finite complement topology — Munkres §17.14
Equivalents of the T1 axiom — Munkres §17.15
Closures and separation under five topologies on the line — Munkres §17.16
Closures of intervals in finer topologies on the line — Munkres §17.17
Closures in the ordered square — Munkres §17.18
Interior and boundary of a subset — Munkres §17.19
Boundaries and interiors of plane regions — Munkres §17.20
Kuratowski’s fourteen-set theorem — Munkres §17.21

18

Epsilon-delta continuity implies open-set continuity — Munkres §18.1
Continuous images of limit points — Munkres §18.2
The identity map between two topologies on a set — Munkres §18.3
Coordinate slices are imbeddings — Munkres §18.4
Intervals homeomorphic with the unit interval — Munkres §18.5
A function continuous at exactly one point — Munkres §18.6
Right-continuous functions and the lower limit topology — Munkres §18.7
The set where f is at most g is closed — Munkres §18.8
Pasting over a locally finite closed cover — Munkres §18.9
Products of continuous maps are continuous — Munkres §18.10
Continuity implies separate continuity — Munkres §18.11
Separate continuity does not imply continuity — Munkres §18.12
Uniqueness of continuous extensions into Hausdorff spaces — Munkres §18.13

19

Basis theorem for the box and product topologies — Munkres §19.1
Products of subspaces are subspaces — Munkres §19.2
Products of Hausdorff spaces in the box and product topologies — Munkres §19.3
Associativity of finite products — Munkres §19.4
The box topology implication in the maps-into-products theorem — Munkres §19.5
Convergence in products via coordinate convergence — Munkres §19.6
Closure of the eventually-zero sequences in box and product topologies — Munkres §19.7
An affine coordinate homeomorphism of R^omega — Munkres §19.8
Choice axiom and nonempty cartesian products — Munkres §19.9
The coarsest topology making a family of maps continuous — Munkres §19.10

20

\(\ell^1\) and \(\ell^p\) metrics induce the usual topology on \(\RR^n\) — Munkres §20.1
The dictionary order plane is metrizable — Munkres §20.2
The metric is continuous exactly for its own topology — Munkres §20.3
Continuity and convergence in the product, uniform, and box topologies — Munkres §20.4
Closure of the eventually-zero sequences in the uniform topology — Munkres §20.5
Uniform balls are not products of intervals — Munkres §20.6
Continuity of the affine map in the uniform topology — Munkres §20.7
Four topologies on \(\ell^2\) and the Hilbert cube — Munkres §20.8
The euclidean metric via the Cauchy-Schwarz inequality — Munkres §20.9
\(\ell^2\) is a vector space with the \(\ell^2\) metric — Munkres §20.10
A bounded metric giving the same topology — Munkres §20.11

21

The subspace metric — Munkres §21.1
Isometric imbeddings — Munkres §21.2
Product metrics, finite and countable — Munkres §21.3
First countability of the lower limit topology and the ordered square — Munkres §21.4
Arithmetic of convergent sequences — Munkres §21.5
Powers converge pointwise but not uniformly on \([0,1]\) — Munkres §21.6
Uniform convergence as convergence in the uniform metric — Munkres §21.7
Interchanging limits under uniform convergence — Munkres §21.8
Pointwise but not uniform convergence to the zero function — Munkres §21.9
Closed sets via the closed-set formulation of continuity — Munkres §21.10
Standard facts about infinite series and the Weierstrass M-test — Munkres §21.11
Continuity of the field operations on \(\RR\) — Munkres §21.12

22

Quotient topology on a three-point image of the line — Munkres §22.1
Continuous maps with continuous sections are quotient maps — Munkres §22.2
A quotient map that is neither open nor closed — Munkres §22.3
Quotients of the plane by level-set relations — Munkres §22.4
Restrictions of open maps to open subspaces are open — Munkres §22.5
Collapsing K in the K-topology — Munkres §22.6
Examples of topological groups — Munkres §22 Supplementary
Continuity of \(xy^{-1}\) characterizes topological groups — Munkres §22 Supplementary
Subgroups and closures of subgroups are topological groups — Munkres §22 Supplementary
Translations are homeomorphisms; topological groups are homogeneous — Munkres §22 Supplementary
Coset spaces of topological groups — Munkres §22 Supplementary
\(\RR/\ZZ\) as a familiar topological group — Munkres §22 Supplementary
Separation properties of topological groups and coset spaces — Munkres §22 Supplementary

23

Connectedness under refinement of topologies — Munkres §23.1
Unions of intersecting sequences of connected subspaces — Munkres §23.2
Unions of connected sets meeting a common connected set — Munkres §23.3
Infinite sets are connected in the finite complement topology — Munkres §23.4
Discrete spaces are totally disconnected — Munkres §23.5
Connected sets crossing a set and its complement meet the boundary — Munkres §23.6
The lower limit line is disconnected — Munkres §23.7
Connectedness of \(\RR^\omega\) in the uniform topology — Munkres §23.8
Punctured products of connected spaces are connected — Munkres §23.9
Arbitrary products of connected spaces are connected — Munkres §23.10
Connected fibers over a connected base connect the total space — Munkres §23.11
Union with the removed subspace across a separation — Munkres §23.12

24

No two of the unit interval variants are homeomorphic — Munkres §24.1
Antipodal points with equal values on the circle — Munkres §24.2
Fixed points of continuous self-maps of the interval — Munkres §24.3
Connected ordered sets are linear continua — Munkres §24.4
Interior and boundary of a connected set — Munkres §24.11
Well-ordered sets times the half-open interval are linear continua — Munkres §24.6
Order-preserving surjections are homeomorphisms — Munkres §24.7
Behavior of path connectedness under products, closures, images, and unions — Munkres §24.8
The plane minus countably many points is path connected — Munkres §24.9
Open connected subsets of the plane are path connected — Munkres §24.10
Linear continua in the dictionary order — Munkres §24.5
The long line — Munkres §24.12

25

Components and continuous maps out of the lower limit line — Munkres §25.1
Components of R^omega in product, uniform, and box topologies — Munkres §25.2
The ordered square is locally connected but not locally path connected — Munkres §25.3
Connected open sets in locally path connected spaces — Munkres §25.4
A rational fan and a nowhere locally connected path-connected set — Munkres §25.5
Weak local connectedness implies local connectedness — Munkres §25.6
Weak local connectedness at a point without local connectedness — Munkres §25.7
Quotients of locally connected spaces are locally connected — Munkres §25.8
The identity component of a topological group is normal — Munkres §25.9
Quasicomponents — Munkres §25.10

26

Compactness under comparable topologies — Munkres §26.1
Compact subspaces in the finite complement and countable complement topologies — Munkres §26.2
Finite unions of compact subspaces are compact — Munkres §26.3
Compact subspaces of metric spaces are closed and bounded — Munkres §26.4
Separating disjoint compact sets in a Hausdorff space — Munkres §26.5
Maps from compact to Hausdorff spaces are closed — Munkres §26.6
Projections with compact factor are closed maps — Munkres §26.7
Closed graph theorem for compact Hausdorff ranges — Munkres §26.8
Generalized tube lemma for compact rectangles — Munkres §26.9
Monotone pointwise convergence to a continuous limit is uniform — Munkres §26.10
Intersections of nested families of closed connected sets are connected — Munkres §26.11
Perfect maps pull compactness back — Munkres §26.12
Products of compact sets in topological groups — Munkres §26.13

27

Compact closed intervals imply the least upper bound property — Munkres §27.1
Distance to a set in a metric space — Munkres §27.2
Compactness, connectedness, and path connectedness of the K-topology line — Munkres §27.3
Connected metric spaces with more than one point are uncountable — Munkres §27.4
Countable unions of nowhere dense closed sets in compact Hausdorff spaces — Munkres §27.5
The Cantor set — Munkres §27.6

28

An infinite set without limit points in the uniform topology — Munkres §28.1
The unit interval is not limit point compact in the lower limit topology — Munkres §28.2
Behavior of limit point compactness under images, closed subsets, and Hausdorff ambient spaces — Munkres §28.3
Countable compactness equals limit point compactness for T1 spaces — Munkres §28.4
Nested closed sets in countably compact spaces — Munkres §28.5
Isometries of compact metric spaces are surjective — Munkres §28.6
Shrinking maps and contractions on compact metric spaces — Munkres §28.7
Compactness, limit point compactness, and sequential compactness in second-countable Hausdorff or metric spaces
Compactness, limit point compactness, and sequential compactness are equivalent for metrizable spaces

29

The rationals are not locally compact — Munkres §29.1
Local compactness of products — Munkres §29.2
Images of locally compact spaces — Munkres §29.3
The Hilbert cube is not locally compact in the uniform topology — Munkres §29.4
Homeomorphisms extend to one-point compactifications — Munkres §29.5
The one-point compactification of R is the circle — Munkres §29.6
The one-point compactification of the minimal uncountable well-ordered set — Munkres §29.7
The one-point compactification of the positive integers — Munkres §29.8
Coset spaces of locally compact groups are locally compact — Munkres §29.9
Compact closure neighborhoods in locally compact Hausdorff spaces — Munkres §29.10
Products of quotient maps with locally compact Hausdorff factors — Munkres §29.11
Examples of directed sets — Munkres §29 Supplementary
Cofinal subsets of directed sets — Munkres §29 Supplementary
Net convergence generalizes sequence convergence — Munkres §29 Supplementary
Coordinatewise convergence of nets in products — Munkres §29 Supplementary
Nets in Hausdorff spaces have at most one limit — Munkres §29 Supplementary
Closure points detected by convergent nets — Munkres §29 Supplementary
Continuity characterized by nets — Munkres §29 Supplementary
Subnets of convergent nets converge — Munkres §29 Supplementary
Accumulation points of nets — Munkres §29 Supplementary
Compactness characterized by subnets — Munkres §29 Supplementary
Closed times compact in topological groups via nets — Munkres §29 Supplementary
Directed sets without antisymmetry — Munkres §29 Supplementary

30

One-point sets are G-delta in first-countable T1 spaces — Munkres §30.1
Every basis of a second-countable space contains a countable basis — Munkres §30.2
Uncountably many limit points of an uncountable set in a second-countable space — Munkres §30.3
Compact metrizable spaces are second countable — Munkres §30.4
Separability and Lindelofness give second countability for metrizable spaces — Munkres §30.5
The lower limit line and the ordered square are not metrizable — Munkres §30.6
Countability axioms of the minimal uncountable well-ordered set — Munkres §30.7
Countability axioms of R^omega in the uniform topology — Munkres §30.8
Closed subspaces of Lindelof spaces; separability is not hereditary — Munkres §30.9
Countable products of separable spaces are separable — Munkres §30.10
Lindelofness and separability pass to continuous images — Munkres §30.11
Countability axioms pass through open maps — Munkres §30.12
Disjoint open collections in separable spaces are countable — Munkres §30.13
Products with a compact factor are Lindelof — Munkres §30.14
Continuous functions on the interval are separable — Munkres §30.15
Density of uncountable products of the line — Munkres §30.16
Countability axioms of the rationals in the box topology — Munkres §30.17
First-countable separable or Lindelof topological groups are second countable — Munkres §30.18
Second countable spaces are first countable — Munkres §30

31

Regular spaces have disjoint closure neighborhoods of points — Munkres §31.1
Normal spaces have disjoint closure neighborhoods of closed sets — Munkres §31.2
Every order topology is regular — Munkres §31.3
Separation axioms under refinement of topologies — Munkres §31.4
The equalizer of two maps into a Hausdorff space is closed — Munkres §31.5
Closed continuous surjections preserve normality — Munkres §31.6
Perfect maps transfer separation and countability properties — Munkres §31.7
Orbit spaces of compact group actions inherit separation properties — Munkres §31.8
Rational and irrational anti-diagonals in the Sorgenfrey plane — Munkres §31.9

32

Closed subspaces of normal spaces are normal — Munkres §32.1
Separation properties project from products to factors — Munkres §32.2
Locally compact Hausdorff spaces are regular — Munkres §32.3
Regular Lindelof spaces are normal — Munkres §32.4
Normality of R^omega in the product and uniform topologies — Munkres §32.5
Complete normality characterized by separated sets — Munkres §32.6
Complete normality of standard spaces and constructions — Munkres §32.7
Every linear continuum is normal — Munkres §32.8
Uncountable powers of the line are not normal — Munkres §32.9
Normality of topological groups — Munkres §32.10

33

Level sets of the Urysohn function — Munkres §33.1
Connected normal and regular spaces with more than one point are uncountable — Munkres §33.2
Direct proof of the Urysohn lemma for metric spaces — Munkres §33.3
Functions vanishing precisely on a closed G-delta set — Munkres §33.4
Strong form of the Urysohn lemma — Munkres §33.5
Perfectly normal spaces — Munkres §33.6
Locally compact Hausdorff spaces are completely regular — Munkres §33.7
Separating compact sets from closed sets in completely regular spaces — Munkres §33.8
Box products of the line are completely regular — Munkres §33.9
Every topological group is completely regular — Munkres §33.10
A regular space that is not completely regular — Munkres §33.11

34

A Hausdorff second-countable space that is not metrizable — Munkres §34.1
Strong hypotheses short of metrizability — Munkres §34.2
Compact Hausdorff spaces are metrizable exactly when second countable — Munkres §34.3
Countable basis versus metrizability for locally compact Hausdorff spaces — Munkres §34.4
Metrizability of the one-point compactification — Munkres §34.5
Details of the imbedding theorem proof — Munkres §34.6
Locally metrizable compact Hausdorff spaces are metrizable — Munkres §34.7
Locally metrizable regular Lindelof spaces are metrizable — Munkres §34.8
Compact unions of closed metrizable subspaces are metrizable — Munkres §34.9

35

The Tietze extension theorem implies the Urysohn lemma — Munkres §35.1
The three-way split in the Tietze theorem proof — Munkres §35.2
Bounded under every metric, bounded continuous functions, and limit point compactness — Munkres §35.3
Retracts of Hausdorff spaces and the plane — Munkres §35.4
The universal extension property — Munkres §35.5
Absolute retracts — Munkres §35.6
The logarithmic spiral and the knotted x-axis as retracts — Munkres §35.7
Absolute retracts are exactly the spaces with the universal extension property — Munkres §35.8
Topology coherent with an increasing sequence of closed subspaces — Munkres §35.9

36

Manifolds are regular and hence metrizable — Munkres §36.1
Compact locally imbeddable spaces imbed in euclidean space — Munkres §36.2
Compact locally euclidean Hausdorff spaces are manifolds — Munkres §36.3
The shrinking lemma for point-finite open coverings — Munkres §36.4
The line with two origins — Munkres §36.5
Properties of eleven standard spaces — Munkres §36 Supplementary
Properties every metric space has — Munkres §36 Supplementary
Properties of compact Hausdorff spaces — Munkres §36 Supplementary
Heredity of the seventeen basic properties — Munkres §36 Supplementary
The seventeen basic properties under products — Munkres §36 Supplementary
The seventeen basic properties under continuous maps — Munkres §36 Supplementary
Paracompactness and topological completeness in the basics review — Munkres §36 Supplementary

37

Maximal collections with the finite intersection property — Munkres §37.1
The countable intersection property characterizes Lindelof spaces — Munkres §37.2
Where the countable intersection property breaks the Tychonoff argument — Munkres §37.3
Quasicomponents equal components in compact Hausdorff spaces — Munkres §37.4
The Tychonoff theorem via the well-ordering theorem — Munkres §37.5

38

Verifying the one-point compactification example for (0,1) — Munkres §38.1
Extending \(\cos(1/x)\) by enlarging the compactification — Munkres §38.2
Metrizable spaces with metrizable compactifications — Munkres §38.3
The Stone–Čech compactification is maximal among compactifications — Munkres §38.4
Compactifications of the minimal uncountable well-ordered set — Munkres §38.5
\(X\) is connected if and only if \(\beta(X)\) is connected — Munkres §38.6
The Stone–Čech compactification of a discrete space is extremally disconnected — Munkres §38.7
Cardinality of the Stone–Čech compactification of \(\ZZ_+\) — Munkres §38.8
Nonmetrizability of the Stone–Čech compactification — Munkres §38.9
The Stone–Čech compactification is a functor — Munkres §38.10

39

Locally finite interval collections in \(\RR\) and in \((0,1)\) — Munkres §39.1
A point-finite open covering that is not locally finite — Munkres §39.2
Closures of a non-locally-finite collection may be locally finite — Munkres §39.3
Refinements among four interval collections — Munkres §39.4
Countably locally finite equals countable in second-countable spaces — Munkres §39.5
A countably locally finite collection that is neither countable nor locally finite — Munkres §39.6

40

\(G_\delta\) sets in first-countable Hausdorff spaces, \(\overline{S}_\Omega\), and metric spaces — Munkres §40.1
Complements of \(F_\sigma\) sets are \(G_\delta\) sets — Munkres §40.2
A \(T_1\) space with a locally finite basis is discrete — Munkres §40.3
A nondiscrete space with a countably locally finite basis and no countable basis — Munkres §40.4
The Bing metrization theorem — Munkres §40.5

41

Paracompact coverings need not have locally finite subcoverings — Munkres §41.1
Products with a compact factor are paracompact — Munkres §41.2
Local compactness does not imply paracompactness — Munkres §41.3
Discrete spaces are paracompact; images need not be — Munkres §41.4
The expansion lemma for locally finite families — Munkres §41.5
\(\sigma\)-compact regular spaces are paracompact — Munkres §41.6
Unions of closed paracompact subspaces — Munkres §41.7
Perfect maps and paracompactness — Munkres §41.8
Locally compact connected topological groups are paracompact — Munkres §41.9
Components of locally compact paracompact Hausdorff spaces are second countable — Munkres §41.10

42

Local metrizability criteria as cases of the Smirnov metrization theorem — Munkres §42.1
Sections of the minimal uncountable well-ordered set are second countable — Munkres §42.2

43

Local compactness of balls versus completeness — Munkres §43.1
Uniformly continuous maps extend to the closure — Munkres §43.2
Metrically equivalent metrics and completeness — Munkres §43.3
Completeness via nested closed sets of vanishing diameter — Munkres §43.4
Banach fixed point theorem for contractions of complete metric spaces — Munkres §43.5
Topologically complete spaces — Munkres §43.6
\(\ell^2\) is complete — Munkres §43.7
Continuity of the evaluation map — Munkres §43.8
Completion via Cauchy sequences — Munkres §43.9
Uniqueness of the completion — Munkres §43.10

44

Continuous surjections from the interval onto cubes — Munkres §44.1
Continuous surjections from \(\RR\) onto \(\RR^n\) — Munkres §44.2
Surjections from \(\RR\) onto \(\RR^\omega\) and \(\RR^\infty\) under the product, uniform, and box topologies — Munkres §44.3
Peano spaces and the Hahn-Mazurkiewicz theorem — Munkres §44.4

45

Countable products of compact metrizable spaces are compact — Munkres §45.1
Finite families, uniform limits, and families with bounded derivatives are equicontinuous — Munkres §45.2
Arzelà’s theorem — Munkres §45.3
Pointwise bounded collections that fail equicontinuity — Munkres §45.4
Compact subsets of functions vanishing at infinity — Munkres §45.5
Ascoli’s theorem over proper metric targets — Munkres §45.6
The Hausdorff metric on closed bounded subsets — Munkres §45.7
Graphs as an imbedding of the function space into the hyperspace — Munkres §45.8

46

The topology of compact convergence — Munkres §46.1
Uniform, compact-convergence, and pointwise topologies on \(Y^X\) — Munkres §46.2
Bounded functions under uniform and compact convergence topologies — Munkres §46.3
Convergence of \(x/n\) in the pointwise, compact-convergence, and uniform topologies — Munkres §46.4
Partial sums of \(\sum k x^k\) converge compactly but not uniformly on \((-1,1)\) — Munkres §46.5
Separation properties of the compact-open topology — Munkres §46.6
Continuity of composition in the compact-open topology — Munkres §46.7
Continuity of evaluation forces the compact-open topology — Munkres §46.8
Products with a locally compact Hausdorff factor preserve quotient maps — Munkres §46.9
Completeness of the compact convergence topology over \(\sigma\)-compact domains — Munkres §46.10
The fine topology on function spaces — Munkres §46.11

47

Pointwise boundedness and equicontinuity of four collections — Munkres §47.1
Compact closures in the topology of compact convergence — Munkres §47.2
The general Ascoli theorem implies the classical version — Munkres §47.3
Arzelà’s theorem over \(\sigma\)-compact Hausdorff domains — Munkres §47.4
Equicontinuous pointwise limits converge compactly — Munkres §47.5

48

Countable unions in Baire spaces — Munkres §48.1
The line as a countable union of sets with empty interior — Munkres §48.2
Locally compact Hausdorff spaces are Baire spaces — Munkres §48.3
Local Baireness implies Baireness — Munkres §48.4
\(G_\delta\) subspaces of compact Hausdorff or complete metric spaces are Baire — Munkres §48.5
The irrationals are a Baire space — Munkres §48.6
Continuity sets are \(G_\delta\); no function is continuous exactly on a countable dense set — Munkres §48.7
Pointwise limits of continuous functions on \(\RR\) are continuous at uncountably many points — Munkres §48.8
Thomae-type function continuous at each irrational — Munkres §48.9
Uniform boundedness principle — Munkres §48.10
Baire property of the lower limit line — Munkres §48.11
Powers of the line are Baire spaces in the box, product, and uniform topologies — Munkres §48.12
Function spaces are Baire in the fine topology — Munkres §48.13

49

Secant-quotient bounds for the building blocks of a nowhere-differentiable function — Munkres §49.1
Small functions in the nowhere-differentiability open sets — Munkres §49.2

50

Discrete spaces have dimension zero — Munkres §50.1
Connected \(T_1\) spaces have positive dimension — Munkres §50.2
The topologist’s sine curve has dimension one — Munkres §50.3
Five points in general position and a linear \(K_5\) in \(\RR^3\) — Munkres §50.4
The imbedding theorem for \(m=1\) and linear graphs in \(\RR^3\) — Munkres §50.5
Second countable locally compact Hausdorff spaces of finite dimension imbed as closed subspaces — Munkres §50.6
Every \(m\)-manifold imbeds as a closed subspace of \(\RR^{2m+1}\) — Munkres §50.7
The dimension of a \(\sigma\)-compact Hausdorff space is bounded by that of its compact subspaces — Munkres §50.8
Manifolds have dimension at most m — Munkres §50.9
Closed subspaces of \(\RR^N\) have dimension at most \(N\) — Munkres §50.10
Characterization of closed imbeddings into \(\RR^N\) — Munkres §50.11
Locally euclidean spaces are locally compact and locally metrizable — Munkres §50 Supplementary
The hierarchy of conditions on locally euclidean spaces — Munkres §50 Supplementary
The line is a manifold that is not compact — Munkres §50 Supplementary
The dictionary order plane is locally euclidean and metrizable but not a manifold — Munkres §50 Supplementary
The long line is normal but not metrizable — Munkres §50 Supplementary
The Prüfer manifold — Munkres §50 Supplementary
Hausdorff equals completely regular for locally euclidean spaces — Munkres §50 Supplementary
Metrizability equals paracompact Hausdorff for locally euclidean spaces — Munkres §50 Supplementary
Components of metrizable locally euclidean spaces are manifolds — Munkres §50 Supplementary

51

Homotopy of composites — Munkres §51.1
Homotopy classes into an interval and out of a path-connected space — Munkres §51.2
Contractible spaces — Munkres §51.3

52

Star-convex sets are simply connected — Munkres §52.1
Change of base point along a composite path — Munkres §52.2
Abelian fundamental groups and base-point independence — Munkres §52.3
Retractions induce surjections on fundamental groups — Munkres §52.4
Maps that extend over \(\RR^n\) induce the trivial homomorphism on \(\pi_1\) — Munkres §52.5
Induced homomorphisms up to base-point change — Munkres §52.6
The fundamental group of a topological group is abelian — Munkres §52.7

53

Projection from a product with a discrete space is a covering map — Munkres §53.1
Uniqueness of the slicing over connected evenly covered sets — Munkres §53.2
The number of sheets of a covering over a connected base — Munkres §53.3
Composites of covering maps with finite fibers — Munkres §53.4
The power maps of the circle as coverings — Munkres §53.5
Properties of the total space of a covering — Munkres §53.6

54

Where path lifting fails for local homeomorphisms — Munkres §54.1
The order of the rectangles in the lifting-of-homotopies proof — Munkres §54.2
Liftings of composite paths — Munkres §54.3
Lifting paths in the polar covering of the punctured plane — Munkres §54.4
Lifting a \((1,2)\) loop on the torus to \(\RR^2\) — Munkres §54.5
Induced homomorphisms of the power and reciprocal power maps — Munkres §54.6
The fundamental group of the torus via the circle covering — Munkres §54.7
Coverings over simply connected bases are homeomorphisms — Munkres §54.8

55

Fixed points for maps of retracts of the disk — Munkres §55.1
Nulhomotopic maps of the circle have fixed and antipodal points — Munkres §55.2
Positive eigenvalues for nonnegative matrices — Munkres §55.3
Consequences of the no-retraction theorem in all dimensions — Munkres §55.4

56

Root location from small lower coefficients — Munkres §56.1
A root bound for a specific septic — Munkres §56.2

57

The theorem of meteorology — Munkres §57.1
Antipode-separating maps of the sphere are surjective — Munkres §57.2
Antipode-preserving maps of the circle act by odd powers — Munkres §57.3
Consequences of non-nulhomotopy of antipode-preserving maps in all dimensions — Munkres §57.4

58

Transitivity of deformation retracts — Munkres §58.1
Fundamental groups of twelve standard spaces — Munkres §58.2
Homotopy equivalence is an equivalence relation — Munkres §58.3
The figure eight and the theta space are homotopy equivalent — Munkres §58.4
Contractibility and the one-point homotopy type — Munkres §58.5
Retracts of contractible spaces — Munkres §58.6
When the inclusion induces an isomorphism on fundamental groups — Munkres §58.7
A homotopy equivalence to a point that is not a deformation retract — Munkres §58.8
The degree of a map of the circle — Munkres §58.9
Consequences of an abstract degree theory on spheres — Munkres §58.10

59

Two spheres touching at a point — Munkres §59.1
A flawed proof that the sphere is simply connected — Munkres §59.2
Euclidean spaces of different dimensions are not homeomorphic — Munkres §59.3
Fundamental groups under the Seifert-van Kampen hypotheses with trivial inclusions — Munkres §59.4

60

Fundamental groups of the solid torus and the torus-sphere product — Munkres §60.1
Antipodal identification on the disk gives the projective plane — Munkres §60.2
The two-fold covering of the figure eight restricted to the axes — Munkres §60.3
The projective line and its covering — Munkres §60.4
A two-fold covering detecting nonabelian fundamental group — Munkres §60.5

61

Simple closed curves on the torus may or may not separate — Munkres §61.1
The closed topologist’s sine curve separates the sphere — Munkres §61.2

62

The Borsuk lemma fails without injectivity — Munkres §62.1
Compact contractible subsets of the sphere do not separate — Munkres §62.2
Extension of homotopic maps into open euclidean sets — Munkres §62.3
Winding of a curve about the origin from its position in the complement — Munkres §62.4
Simply connected open sets contain the interiors of their curves — Munkres §62.5
Higher-dimensional separation theorems from the no-retraction theorem — Munkres §62.6

63

Two disjoint curves on the sphere make three regions — Munkres §63.1
Attaching arcs and curves to a separating set on the sphere — Munkres §63.2
Separation by topologist’s sine curve variants on the sphere — Munkres §63.3

64

Hausdorffness of finite unions of arcs — Munkres §64.1

66

The winding number as a degree — Munkres §66.1
Winding number one for smooth simple closed curves — Munkres §66.2

67

Directness of sums via vanishing of finite sums — Munkres §67.1
Subgroups need not be direct summands — Munkres §67.2
A change of basis in a free abelian group — Munkres §67.3
Torsion subgroups and the rationals — Munkres §67.4
A full-rank proper subgroup of a free abelian group — Munkres §67.5
Subgroups of free abelian groups are free of no greater rank — Munkres §67.6

68

Free product construction in §68 — Munkres §68.1
Word length in free products — Munkres §68.2
Conjugacy classes of finite order in a free product — Munkres §68.3
Uniqueness of free products — Munkres §68.4

69

Abelianization of a free product of two groups — Munkres §69.1
Abelianization of arbitrary free products — Munkres §69.2
Free products of finite cyclic groups determine their factors — Munkres §69.3
Direct sums of cyclic groups are not determined by their factors — Munkres §69.4

70

Seifert-van Kampen with trivial intersection homomorphism — Munkres §70.1
Seifert-van Kampen when one inclusion is surjective — Munkres §70.2
Finite presentations of free products and pushouts — Munkres §70.3

71

Hausdorffness of wedges of circles — Munkres §71.1
Fundamental groups of wedges with nice neighborhoods — Munkres §71.2
Fundamental group of the wedge of a circle and a sphere — Munkres §71.3
Infinite wedges of circles are not first countable — Munkres §71.4
The countable chain of tangent circles — Munkres §71.5

72

Adjoining cells of dimension above two does not change the fundamental group — Munkres §72.1
Adjunction spaces satisfy the hypotheses for adjoining a two-cell — Munkres §72.2
Quotienting the fundamental group by adjoining a two-cell — Munkres §72.3

73

Realizing finitely generated abelian groups and cyclic free products — Munkres §73.1
Finitely presented groups as fundamental groups of compact Hausdorff spaces — Munkres §73.2
Kernels under surjective composites of homomorphisms — Munkres §73.3
The two-dimensional CW complex is metrizable — Munkres §73.4

74

A presentation for the fundamental group of the connected sum of the projective plane and the torus — Munkres §74.1
A seven-sided scheme whose group is a free product of cyclics — Munkres §74.2
The Klein bottle’s fundamental group and its torus double cover — Munkres §74.3
The Klein bottle as a connected sum of two projective planes — Munkres §74.4
The Mobius band is a punctured projective plane — Munkres §74.5
Nonabelian fundamental groups of higher-genus tori — Munkres §74.6
Nonabelian fundamental groups of higher projective planes — Munkres §74.7

75

The first homology of the connected sum of the projective plane and the torus — Munkres §75.1
The first homology of the Klein bottle — Munkres §75.2
An eight-sided scheme with a single vertex class — Munkres §75.3
An eight-sided scheme with several vertex classes — Munkres §75.4

76

Pasting schemes for two polygonal regions — Munkres §76.1
Two reduction sequences for the same scheme disagree — Munkres §76.2

77

Normal forms for pasting schemes — Munkres §77.1
Elementary operations behind the classification figures — Munkres §77.2
The reduction algorithm applied to eight schemes — Munkres §77.3
Reducing schemes of ten sides to standard form — Munkres §77.4

78

Triangular schemes for two collections of four triangles — Munkres §78.1
The boundary of a surface with boundary is intrinsic — Munkres §78.2
The closed disk as a surface with boundary — Munkres §78.3
Surfaces with k holes — Munkres §78.4
Classification of compact connected triangulable surfaces with boundary — Munkres §78.5

79

Maps of higher spheres into the circle are nulhomotopic — Munkres §79.1
Maps of the projective plane and the torus into the circle — Munkres §79.2
Normal subgroups and regular coverings — Munkres §79.3
Coverings of the torus from subgroups of \(\ZZ\times\ZZ\) — Munkres §79.4
Coverings of the torus are classified by rank — Munkres §79.5
Covering maps of topological groups lift the group structure — Munkres §79.6
Covering homomorphisms of abelian topological groups — Munkres §79.7

80

Composites of covering maps over a base with a universal covering — Munkres §80.1

81

Order-two group actions on the torus — Munkres §81.1
Covering transformations of four coverings of the figure eight — Munkres §81.2
Quotients by covering transformation groups — Munkres §81.3
Finite fixed-point-free actions are properly discontinuous — Munkres §81.4
Lens spaces — Munkres §81.5
Properly discontinuous actions with compactly supported interference — Munkres §81.6

82

Simple connectedness implies semilocal simple connectedness — Munkres §82.1
The cone on the infinite earring — Munkres §82.2
Star-fine refinements of open coverings — Munkres §82 Supplementary
Countability of the fundamental group under countable basis and regularity — Munkres §82 Supplementary
Coverings of second-countable regular spaces — Munkres §82 Supplementary
Finite generation of the fundamental group of a compact Hausdorff space — Munkres §82 Supplementary
The uncountable fundamental group of the infinite earring — Munkres §82 Supplementary

83

The vertex assumption in the normality proof for linear graphs — Munkres §83.1
Euler numbers of standard graphs and their coverings — Munkres §83.2

84

Finiteness in the maximal tree argument — Munkres §84.1
Free ranks of complete and utilities graphs — Munkres §84.2
Subgroups of the free group on two generators from four coverings — Munkres §84.3

85

The Euler number is a topological invariant — Munkres §85.1
A cyclic subgroup of infinite index in the free group of rank two — Munkres §85.2
Free generators for the subgroup lifted over the figure eight — Munkres §85.3

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