Appendix E Examples
0.1 Sets and functions
Example 0.1.11 Power set
Example 0.1.19 Arithmetic operations as functions
Example 0.1.25 Role of domain and codomain in injectivity and surjectivity
0.2 Logic
Example 0.2.7 Modeling “Every positive number has a square-root”
Example 0.2.10 The limit does not exist
0.3 Proof techniques
Example 0.3.3 Proof: invertible is equivalent to bijective
Example 0.3.4 Proof by contradiction
Example 0.3.8 Weak induction
Example 0.3.10 Strong induction
1.1 Topological spaces
Example 1.1.3 Some small topological spaces
Example 1.1.5 Comparing trivial, discrete, cofinite topologies
1.2 Topological basis
Example 1.2.4 Some bases for \(\R\)
Example 1.2.8 A basis for the standard topology on \(\R\)
Example 1.2.9 Multiple bases for the standard real topology
Example 1.2.10 Lower limit is finer than standard
Example 1.2.11 Comparing standard, lower limit, and \(K\)-topology
Example 1.2.13 Bases for discrete topology
1.3 Metric spaces
Example 1.3.4 Metric balls for Euclidean and box metrics
Example 1.3.7 Equivalence of Euclidean, box, and taxicab metrics
Example 1.3.9 Trivial metric
1.4 Closed sets, closure, and interior
Example 1.4.6 Closed in trivial and discrete topologies
Example 1.4.7 Closed in the cofinite topology
Example 1.4.9 Closed sets in Euclidean metric topology
Example 1.4.13 Interior and closure of \(K\)
Example 1.4.14 Interior and closure in cofinite topology
1.5 Limit points and the Hausdorff property
Example 1.5.3 Limit points in \(\R\)
Example 1.5.11 Metric spaces are Hausdorff
Example 1.5.13 Limits in the cofinite topology
1.6 Subspaces and finite products
Example 1.6.2 Open/closed depends on topology
Example 1.6.4 Product basis not a topology necessarily
1.7 Arbitrary products
Example 1.7.3 Infinite binary sequences
1.8 Convergence in product spaces
Example 1.8.5 Convergent sequences in \(\R^\omega\)
1.9 Continuous functions
Example 1.9.3 Continuous functions: basic examples
Example 1.9.4 Non-continuous inclusion
Example 1.9.5 Products and projections
1.10 Homeomorphisms
Example 1.10.3 Homeomorphism: inverse must be continuous
Example 1.10.5 All open intervals of \(\R\) are homeomorphic
Example 1.10.10 Topological properties
Example 1.10.15 Polar transformation
Example 1.10.16 Continuous bijection onto circle
1.11 Quotients
Example 1.11.11 Quotient by an equivalence relation
Example 1.11.12 Quotient of product space
Example 1.11.13 Line with doubled origin
Example 1.11.14 The circle as a quotient space
1.12 Connected spaces
Example 1.12.3 Examples
1.13 Path-connected spaces
Example 1.13.5 Balls in \(\R^n\)
Example 1.13.8 The \(n\)-sphere
1.14 Compact spaces
Example 1.14.4 Elementary examples
Example 1.14.5 \(K\cup\{0\}\) is compact
1.15 Compactness in \(\R^n\)
Example 1.15.3 Boxes and balls in \(\R^n\)
1.16 Compactness in metric spaces
1.17 Locally compact spaces and compactification
Example 1.17.8 One-point compactification of \(\R\)
1.18 Countability axioms
Example 1.18.2 Metric spaces are first countable
Example 1.18.3 First countable cofinite spaces
Example 1.18.12 \(\R^n\) is second countable
Example 1.18.13 \(\R_\ell\) is not metrizable
Example 1.18.15 First countable: continuous image
Example 1.18.16 \(\R^\R\) is not metrizable.
1.19 Regular and normal spaces
Example 1.19.5 Elementary examples
Example 1.19.7 \(\R_K\) is not regular
Example 1.19.11 \(\R_\ell\) is regular
Example 1.19.15 \(\R_\ell\) is normal
Example 1.19.16 \(\R_\ell\times \R_\ell\) is not normal
1.22 Nets
Example 1.22.2 Common examples
Example 1.22.4 Common examples
Example 1.22.6 Riemann integral
2.1 Homotopy
Example 2.1.4 Homotopic functions to \(\R^n\)
Example 2.1.7 Homotopic paths in \(\R^n\)
Example 2.1.8 Paths with non-convex codomain
2.3 Covering spaces
Example 2.3.2 Trivial covering
Example 2.3.3 Covering map \(\R\rightarrow S^1\)
Example 2.3.6 Covering of \(\mathbb{P}^2\)
Example 2.3.8 Not all quotient maps are covering maps
Example 2.3.9 Local homeomorphism not sufficient
Example 2.3.11 Covering of torus
Example 2.3.13 Covering of figure eight
2.4 Lifting correspondence
Example 2.4.6 Fundamental group of \(S^1\)
Example 2.4.7 Fundamental group of \(T=S^1\times S^1\)
2.5 Retractions and Brouwer fixed point
2.6 Deformation retract
Example 2.6.6 \(\R^3\) minus axis
Example 2.6.8 Doubly punctured plane, figure eight, theta space
2.7 Homotopty equivalence
Example 2.7.2 Deformation retract
2.9 Fundamental groups of some surfaces
Example 2.9.3 Sphere, projective 2-space, torus
Example 2.9.6 Double torus
Example 2.9.7 Fundamental group of the figure eight
Example 2.9.8 Double torus fundamental group
2.12 Free Abelian groups
2.14 Free groups
Example 2.14.7 Presentations of groups
2.16 Wedge of circles
Example 2.16.2 Wedge of \(n\) cirles
2.17 Adjoining a 2-cell
Example 2.17.4 Torus revisited (again)
2.21 Classification of covering spaces
Example 2.21.11 Coverings of \(S^1\)
