Continuous Functions Review

Algebraic Topology · Axiom Academy

LESSON Continuous Functions Review Understanding continuity in topological spaces through the preimage definition 1. Topological Definition of Continuity Let (X, τ_X) and (Y, τ_Y) be topological spaces. A function f: X → Y is continuous if the preimage of every open set in Y is open in X. f: X → Y is continuous if and only if: For all V ∈ τ_Y (V open in Y), we have f⁻¹(V) ∈ τ_X (f⁻¹(V) open in X) This elegant definition captures the intuition that continuity preserves "nearness" - nearby points map to nearby points - but without reference to distance. 2. Equivalent Characterizations The preimage definition is equivalent to several other useful formulations that sometimes make proofs easier. Equivalent conditions for f: X → Y continuous: 1. f⁻¹(V) is open in X for all open V in Y (definition) 2. f⁻¹(F) is closed in X for all closed F in Y 3. f⁻¹(B) is open in X for all B in a basis for Y 4. f(cl(A)) ⊆ cl(f(A)) for all A ⊆ X 3. Composition of Continuous Functions One of the most important properties of continuous functions is that they compose nicely. If f: X → Y and g: Y → Z are continuous, then the composition g ∘ f: X → Z is also continuous. Proof idea: If W is open in Z, then g⁻¹(W) is open in Y (g continuous), so (g∘f)⁻¹(W) = f⁻¹(g⁻¹(W)) is open in X (f continuous). 4. Key Examples of Continuous Functions Understanding examples helps develop intuition for what continuity means topologically. 1. Identity map id_X: X → X is always continuous

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