Appendix C Theory
0.1 Sets and functions
Theorem 0.1.28 Invertible is equivalent to bijective
1.2 Topological basis
Lemma 1.2.3 Covering principle
Theorem 1.2.5 Topology generated by basis
Theorem 1.2.7 Comparing topologies via basis
Theorem 1.2.12 Basis criterion
1.3 Metric spaces
Lemma 1.3.5 Metric balls
Theorem 1.3.6 Comparing metric topologies
1.4 Closed sets, closure, and interior
Theorem 1.4.2 Property of closed sets
Theorem 1.4.5 Closed sets axioms
Lemma 1.4.8 Basis description of closed sets
Theorem 1.4.11 Equivalent notions of interior and closure
1.5 Limit points and the Hausdorff property
Theorem 1.5.4 Limit points and closure
Corollary 1.5.5 Closed, closure, limit points
Theorem 1.5.7 Hausdorff properties
Corollary 1.5.10 Hausdorff versus \(T_1\)
Theorem 1.5.14 Unique limits in Hausdorff spaces
Theorem 1.5.15 Limit points in \(T_1\)-spaces
1.6 Subspaces and finite products
Theorem 1.6.1 Subspace properties
Theorem 1.6.3 Openness/closedness transitivity
1.7 Arbitrary products
Theorem 1.7.4 Properties of product spaces
1.8 Convergence in product spaces
Theorem 1.8.4 Pointwise convergence and product topology
Theorem 1.8.8 Product topology on \(\R^\omega\)
1.9 Continuous functions
Theorem 1.9.7 Continuity equivalences
Theorem 1.9.9 Building continuous functions
1.10 Homeomorphisms
Theorem 1.10.7 Homeomorphism equivalences
Theorem 1.10.11 Universal mapping property of product space
Corollary 1.10.13 Universal mapping property of product space
Theorem 1.10.17 Universal mapping property of coproduct space
1.11 Quotients
Theorem 1.11.5 Quotient maps and quotient spaces
Theorem 1.11.7 Quotient map properties
Theorem 1.11.10 Universal mapping property of quotient maps
1.12 Connected spaces
Theorem 1.12.4 Connected subsets of \(\R\)
Theorem 1.12.5 Connectedness equivalences
Theorem 1.12.6 Connectedness and subspaces
Theorem 1.12.7 Connectedness and continuity
Corollary 1.12.8 Graphs of continuous functions
1.13 Path-connected spaces
Theorem 1.13.3 Path-connected implies connected
Theorem 1.13.4 Path-connectedness and continuity
Theorem 1.13.10 Connected components
1.14 Compact spaces
Theorem 1.14.6 Finite closed intervals are compact
Theorem 1.14.7 Compact implies closed in Hausdorff spaces
Theorem 1.14.8 Compactness inherited by closed subspaces
Theorem 1.14.9 Compactness preserved under continuous image
Theorem 1.14.10 Compactness and closed maps
Corollary 1.14.11 Compactness and homeomorphisms
Theorem 1.14.13 Closed formulation of compactness
Corollary 1.14.14 Nested closed sets in compact space
1.15 Compactness in \(\R^n\)
Theorem 1.15.1 Tychonoff theorem
Lemma 1.15.2 Tube lemma
Theorem 1.15.5 Heine-Borel theorem
Corollary 1.15.6 Extreme value theorem
Corollary 1.15.8 Lebesgue number
Corollary 1.15.9 Compact: continuous implies uniformly continuous
1.16 Compactness in metric spaces
Theorem 1.16.3 Compactness in metric spaces
Theorem 1.16.5 Compact implies limit point compact
Theorem 1.16.7 Sequential compactness implies limit point compactness
Theorem 1.16.13 Sequentially compact+second countable implies compact
Corollary 1.16.15 Compactness equivalences
Theorem 1.16.16 Metrizable+sequentially compact implies second countable
1.17 Locally compact spaces and compactification
Theorem 1.17.2 Local compactness equivalence
Theorem 1.17.5 One-point compactification
Corollary 1.17.6 One-point compactification
Corollary 1.17.7 Locally compact Hausdorff spaces
1.18 Countability axioms
Theorem 1.18.5 First countable and sequential properties
Theorem 1.18.8 Second countable: strongest countability axiom
Theorem 1.18.14 First/second countable: subspaces, images, and products
1.19 Regular and normal spaces
Theorem 1.19.6 Metric spaces are normal
Theorem 1.19.8 Regular and normal equivalences
Theorem 1.19.9 Locally compact Hausdorff implies regular
Theorem 1.19.13 Regularity: subspace and product properties
Theorem 1.19.17 Compact+Hausdorff \(\implies\) normal
Theorem 1.19.18 Regular+second countable \(\implies\) normal
1.20 Things Urysohn
Theorem 1.20.1 Urysohn lemma
Theorem 1.20.6 Complete regularity: subspace and product properties
Theorem 1.20.7 Urysohn metrization theorem
Theorem 1.20.8 Embedding theorem
1.21 Urysohn offspring
Theorem 1.21.1 Tietze extension theorem
Theorem 1.21.3 Partition of unity (finite)
Corollary 1.21.4 Compact manifold embedding
1.22 Nets
Theorem 1.22.7 Nets and topology
Theorem 1.22.9 Limit points of nets
Lemma 1.22.10 Kelley’s lemma
Theorem 1.22.11 Nets and compactness
1.23 Tychonoff theorem via nets
Theorem 1.23.1 Tychonoff implies AC (Kelley)
Theorem 1.23.3 Zorn’s lemma
Lemma 1.23.4 Nets in products
Theorem 1.23.5 Tychonoff theorem
2.1 Homotopy
Theorem 2.1.9 Homotopy equivalence relation
Theorem 2.1.12 Path product properties
Lemma 2.1.13 Reparametrization of path
Theorem 2.1.14 Partitions of \(I\)
2.2 Fundamental group
Theorem 2.2.4 Fundamental group and path components
Theorem 2.2.6 Fundamental group functor
Corollary 2.2.7 Fundamental group invariance
2.3 Covering spaces
Theorem 2.3.4 Covering map properties
Theorem 2.3.10 Covering map constructions
2.4 Lifting correspondence
Lemma 2.4.2 Uniqueness of liftings
Theorem 2.4.3 Lifting to covering spaces
Theorem 2.4.5 Lifting correspondence
2.5 Retractions and Brouwer fixed point
Theorem 2.5.5 Nullhomotopies from \(S^1\)
Corollary 2.5.6 Some non-nullhomotopies
Theorem 2.5.7 Nonvanishing vector fields on \(B^2\)
Theorem 2.5.8 Brouwer fixed-point theorem (disc)
Corollary 2.5.9 Eigenvalues of positive matrices
2.6 Deformation retract
Lemma 2.6.1 Homotopic maps and fundamental groups
Theorem 2.6.5 Deformation retract and fundamental groups
2.7 Homotopty equivalence
Lemma 2.7.3 Homotopic maps and fundamental group revisited
Corollary 2.7.4 Homotopic maps and fundamental groups
Theorem 2.7.5 Homotopy equivalence and fundamental groups
2.8 Fundamental group of \(S^n\)
Theorem 2.8.1 Weak Seifert-van Kampen
Corollary 2.8.2 Fundamental group of \(S^n\)
Corollary 2.8.4 Fundamental group of \(\PP^n\)
2.9 Fundamental groups of some surfaces
Theorem 2.9.9 Some non-homeomorphic surfaces
2.10 Jordan separation theorem
Theorem 2.10.4 Components in locally connected spaces
Theorem 2.10.8 Jordan separation theorem
2.11 Jordan curve theorem
Theorem 2.11.1 Nonseparation theorem
Theorem 2.11.2 Complementary Seifert-van Kampen
Theorem 2.11.3 Jordan curve theorem
2.12 Free Abelian groups
Theorem 2.12.2 Universal property of quotient groups
Theorem 2.12.4 Universal property of the direct product
Theorem 2.12.7 Universal property of direct sums
Theorem 2.12.10 Direct sum equivalence
Theorem 2.12.12 Free abelian groups
2.13 Free products
Theorem 2.13.2 Free product properties
Theorem 2.13.4 Free products exist
Theorem 2.13.5 Products of free products
Theorem 2.13.7 Quotients of free products
2.14 Free groups
Theorem 2.14.2 Free groups
Theorem 2.14.4 Quotients of free groups
Theorem 2.14.9 Commutator and abelianization
Theorem 2.14.11 Abelianization of free group
Corollary 2.14.12 Free groupRank of free group
2.15 Seifert-van Kampen theorem
Theorem 2.15.2 Pushouts
Theorem 2.15.3 Seifert-van Kampen theorem
Corollary 2.15.4 Seifert-van Kampen: classical
Corollary 2.15.5 Seifert-van Kampen: special cases
2.16 Wedge of circles
Theorem 2.16.3 Wedge of \(n\) circles
Theorem 2.16.5 Wedge of circles
2.17 Adjoining a 2-cell
Theorem 2.17.2 Adjoining a 2-cell
2.18 \(n\)-fold dunce caps
Lemma 2.18.2 Normality of quotients
Theorem 2.18.3 Dunce cap fundamental group
2.19 Pasted polygonal regions
Theorem 2.19.6 Pasted polygonal regions
Theorem 2.19.7 Pasted polygonal fundamental groups
2.20 Classification of surfaces
Theorem 2.20.2 Elementary operations
Theorem 2.20.3 Abelianization of quotients
Theorem 2.20.4 Non-homeomorphic surfaces
Theorem 2.20.5 Classification theorem
2.21 Classification of covering spaces
Theorem 2.21.2 Covering spaces
Lemma 2.21.3 Liftings of coverings
Corollary 2.21.4 Covering spaces
Theorem 2.21.9 Universal covering space
Theorem 2.21.10 Covering space correspondence
