Topological Spaces Review

Algebraic Topology · Axiom Academy

LESSON Topological Spaces Review Revisiting the fundamental definitions and examples of topological spaces 1. Definition of Topological Space A topological space is a pair (X, τ) where X is a set and τ is a collection of subsets of X (called open sets ) satisfying three axioms. 1. The empty set ∅ and X itself are in τ 2. Any union of sets in τ is also in τ (closure under arbitrary unions) 3. Any finite intersection of sets in τ is also in τ (closure under finite intersections) Often we don't specify all open sets directly. Instead, we use a basis to generate the topology. Basis: A collection B of subsets of X is a basis for topology τ if every open set in τ can be written as a union of sets from B. Subbasis: A collection S of subsets is a subbasis if the collection of all finite intersections of sets in S forms a basis. 3. Standard Examples of Topologies Let's review the most important examples of topological spaces that we'll encounter throughout algebraic topology. 1. Standard Topology on ℝⁿ: Generated by open balls 2. Discrete Topology: τ = P(X) (all subsets are open) 3. Indiscrete/Trivial Topology: τ = ∅, X (only ∅ and X are open) 4. Cofinite Topology: Open sets are those whose complement is finite (or empty) 4. Key Properties and Closed Sets Related to open sets are closed sets , which are complements of open sets. They satisfy dual axioms. 2. Any intersection of closed sets is closed 3. Any finite union of closed sets is closed

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