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Appendix B Definitions

0.1 Sets and functions

Definition 0.1.1 Sets

Definition 0.1.2 Set equality

Definition 0.1.3 Set inclusion (subsets)

Definition 0.1.5 Set-builder notation

Definition 0.1.8 Union, intersection, difference, and complement

Definition 0.1.9 Common mathematical sets

Definition 0.1.10 Power set

Definition 0.1.12 Cartesian product (finite)

Definition 0.1.15 I-tuple

Definition 0.1.16 Cartesian product (arbitrary)

Definition 0.1.17 Functions

Definition 0.1.21 Function equality

Definition 0.1.22 Image of a set

Definition 0.1.23 Preimage of set

Definition 0.1.24 Injective, surjective, bijective

Definition 0.1.26 Function composition

Definition 0.1.27 Identity and inverse functions

Definition 0.1.29 Cardinality

0.2 Logic

Definition 0.2.1 Logical operators

Definition 0.2.5

1.1 Topological spaces

Definition 1.1.1 Topological space

Definition 1.1.4 Comparable topologies

1.2 Topological basis

Definition 1.2.1 Topological basis

Definition 1.2.6 Topology generated by basis

1.3 Metric spaces

Definition 1.3.1 Metric space

Definition 1.3.2 Euclidean, box, and taxicab metrics

Definition 1.3.3 Metric balls

Definition 1.3.8 Trivial metric

Definition 1.3.10 \(p\)-adic metric

1.4 Closed sets, closure, and interior

Definition 1.4.1 Closed sets

Definition 1.4.4 Closed sets axioms

Definition 1.4.10 Interior and closure of a set

1.5 Limit points and the Hausdorff property

Definition 1.5.1 Neighborhood of element or set

Definition 1.5.2 Limit point of a set

Definition 1.5.6 Hausdorff property

Definition 1.5.8 The \(T_1\)-axiom

Definition 1.5.12 Convergent sequence

1.7 Arbitrary products

Notation 1.7.2 \(A^\omega\)

1.8 Convergence in product spaces

Definition 1.8.1 Pointwise convergence

Definition 1.8.6 Standard bounded metric

1.9 Continuous functions

Definition 1.9.1 Continuous function

Definition 1.9.8 Continuity at \(x\)

1.10 Homeomorphisms

Definition 1.10.1 Homeomorphism

Definition 1.10.6 Open and closed maps

Definition 1.10.9 Topological properties

Definition 1.10.12 Product of functions

1.11 Quotients

Definition 1.11.1 Quotient by an equivalence relation

Definition 1.11.4 Quotient map

Definition 1.11.6 Saturated sets

Definition 1.11.9 Fibers of maps

1.12 Connected spaces

Definition 1.12.1 Connected space and separations

1.13 Path-connected spaces

Definition 1.13.1 Path

Definition 1.13.2 Path-connected space

Definition 1.13.9 Connected components

1.14 Compact spaces

Definition 1.14.1 Covering

Definition 1.14.2 Compact space

Definition 1.14.12 Finite intersection property

1.15 Compactness in \(\R^n\)

Definition 1.15.4 Bounded, diameter, and distance to sets

Definition 1.15.7 Lebesgue number

1.16 Compactness in metric spaces

Definition 1.16.1 Limit point compact

Definition 1.16.2 Sequentially compact

Definition 1.16.9 Local basis

Definition 1.16.10 First countable

Definition 1.16.12 Second countable

Definition 1.16.17 Perfect space

Definition 1.16.18 Totally disconnected

1.17 Locally compact spaces and compactification

Definition 1.17.1 Locally compact space

Definition 1.17.3 Embedding

Definition 1.17.4 Compactification

1.18 Countability axioms

Definition 1.18.4 Sequentially closed/continuous

Definition 1.18.6 Second countable, Lindelöf, separable

1.19 Regular and normal spaces

Definition 1.19.1 Separated by open sets

Definition 1.19.2 Regular and normal spaces

1.20 Things Urysohn

Definition 1.20.2 Separated by a continuous function

Definition 1.20.4 Completely regular

1.21 Urysohn offspring

Definition 1.21.2 Partition of unity

1.22 Nets

Definition 1.22.1 Partial ordering axioms

Definition 1.22.3 Directed set

Definition 1.22.5 Nets and convergent nets

Definition 1.22.8 Subnets

1.23 Tychonoff theorem via nets

Definition 1.23.2 Upper bounds, maximal elements, chains

2.1 Homotopy

Notation 2.1.1 \(I=[0,1]\)

Definition 2.1.2 Homotopy of maps

Definition 2.1.5 Nullhomotopic map

Definition 2.1.6 Path homotopy

Notation 2.1.10 Homotopy equivalence

Definition 2.1.11 Path product

2.2 Fundamental group

Definition 2.2.1 Group compendium

Definition 2.2.2 Fundamental group

Definition 2.2.3 Simply connected

Definition 2.2.5 Pointed space

2.3 Covering spaces

Definition 2.3.1 Covering map

2.4 Lifting correspondence

Definition 2.4.1 Lifting

Definition 2.4.4 Lifting correspondence

2.5 Retractions and Brouwer fixed point

Definition 2.5.1 Retraction

Definition 2.5.3 \(n\)-Ball

2.6 Deformation retract

Definition 2.6.3 Deformation retract

2.7 Homotopty equivalence

Definition 2.7.1 Homotopy equivalence

2.8 Fundamental group of \(S^n\)

Definition 2.8.3 Projective space

2.9 Fundamental groups of some surfaces

Definition 2.9.1 Topological manifold

Definition 2.9.4 Euclidean balls

Definition 2.9.5 Connected sum of surfaces

2.10 Jordan separation theorem

Definition 2.10.1 Arcs and simple closed curves

Definition 2.10.3

2.12 Free Abelian groups

Definition 2.12.1 \(\Hom(G,H)\)

Definition 2.12.3 Direct product of groups

Definition 2.12.5 Direct sum of abelian groups

Definition 2.12.9 Direct sum of subgroups

Definition 2.12.11 Free abelian group

Definition 2.12.13 Basis of a free abelian group

2.13 Free products

Definition 2.13.1 Free product of groups

Definition 2.13.3 Words and reduced words

Definition 2.13.6 Least normal subgroup

2.14 Free groups

Definition 2.14.1 Free group

Definition 2.14.3 Freely generated

Definition 2.14.5 Group presentation

Definition 2.14.8 Commutator subgroup

Definition 2.14.10 Abelianization

Definition 2.14.13 Rank of free group

2.15 Seifert-van Kampen theorem

Definition 2.15.1 Pushouts of groups

2.16 Wedge of circles

Definition 2.16.1 Wedge of \(n\) circles

Definition 2.16.4 Wedge of circles

2.17 Adjoining a 2-cell

Definition 2.17.1 2-cell

Definition 2.17.3 Adjoining a 2-cell

2.18 \(n\)-fold dunce caps

Definition 2.18.1 \(n\)-fold dunce cap

2.19 Pasted polygonal regions

Definition 2.19.1 Polygonal region

Definition 2.19.2 Oriented line segment

Definition 2.19.4 Oriented labelling

Definition 2.19.5 Pasted polygonal region

Definition 2.19.8 \(n\)-fold torus

Definition 2.19.9 \(m\)-fold projective plane

2.20 Classification of surfaces

Definition 2.20.1 Equivalent labelling schemes

2.21 Classification of covering spaces

Definition 2.21.1 Maps of coverings

Definition 2.21.5 Universal covering space

Definition 2.21.8 Semilocally simply connected