Topology Review Summary

Algebraic Topology · Axiom Academy

SUMMARY Topology Review Summary Essential point-set topology foundations for algebraic topology: key definitions, important spaces, and fundamental concepts that prepare us for studying fundamental groups and homology. Definition: A topological space consists of a set and a collection of subsets (open sets) satisfying: , closure under arbitrary unions, and closure under finite intersections Open vs Closed Sets: Sets in are open; complements of open sets are closed. Sets can be both, neither, or one but not the other Basis: A collection such that every open set is a union of basis elements. Generates the topology and simplifies verification Why It Matters: Topologies formalize the notion of "nearness" and "continuity" without requiring a metric, enabling study of geometric properties invariant under continuous deformations Continuous Functions: is continuous if the preimage of every open set in is open in . This generalizes continuity from analysis Homeomorphism: A bijective continuous function with continuous inverse. Two spaces are homeomorphic if there exists a homeomorphism between them Topological Property: A property preserved by homeomorphisms (e.g., compactness, connectedness). Algebraic topology seeks computable topological invariants Why It Matters: Homeomorphisms capture when two spaces are "the same" topologically. Distinguishing non-homeomorphic spaces motivates algebraic invariants like fundamental groups

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