Topics
Topological spaces and continuous functions
- Continuous functions
- Topological spaces and open sets
- Product spaces
- Hausdorff spaces
- Metrics, metric spaces, and the metric topology
- Quotient spaces
- Disjoint unions
- Universal properties of topologies
Theorems:
- (Lemma) Obtaining a basis from a topology
- (Theorem) Describing the closure of a set using a topological basis
- (Theorem) The closure of a set is the union of the set with its limit points
- (Theorem) Sequences in Hausdorff spaces converge to at most one point
- (Theorem) Continuity of maps relative to the product topology
- (Theorem) The uniform topology is between the box and product topologies
- (Theorem) The countable product of the real line is metrizable
- (Theorem) The limit of a uniformly convergent sequence of continuous functions is continuous
Examples:
- Discrete topology
- Finite complement topology
- The countably infinite product of the real line with itself
Connectedness and compactness
Theorems:
- (Theorem) Every compact metric space is complete
- (Theorem) A closed subspace of a complete metric space is complete
- (Theorem) Tube lemma
- (Theorem) Compactness, limit point compactness, and sequential compactness are equivalent on metric spaces
- (Theorem) Compact subspaces of Hausdorff spaces are closed
- (Theorem) A continuous bijection from a compact space to a Hausdorff space is a homeomorphism
- (Theorem) One-point compactification
- (Lemma) Lebesgue numbers exist for compact metric spaces
Examples:
Countability and separation axioms
Examples:
The fundamental group
- Homotopies
- Fundamental groups
- Covering maps
- Retractions and fixed points
- Contractible spaces
- Deformation retractions
- Homotopy equivalence
- Free products of groups
Theorems:
- (Theorem) The fundamental group of the circle is isomorphic to the additive group of integers
- (Theorem) The fundamental group of the n-sphere is trivial in higher dimensions
- (Theorem) van Kampen
Resources
- -base – database of topological spaces