Product Spaces

Algebraic Topology · Axiom Academy

Product topology, continuous maps on products, and the Tychonoff theorem 1. Product Topology Definition Given topological spaces X and Y , the product topology on X × Y is the topology generated by sets of the form U × V where U is open in X and V is open in Y. The projection maps πₓ: X × Y → X and πᵧ: X × Y → Y are continuous and open The product topology is the coarsest (smallest) topology making both projections continuous Not all open sets are of the form U × V (only basis elements are) 2. Continuous Maps on Products A map into a product space is continuous if and only if both component functions are continuous. This characterizes continuity on products via the universal property . If f: X → X' and g: Y → Y' are continuous, then f × g: X × Y → X' × Y' is continuous The product of two homeomorphisms is a homeomorphism The diagonal map Δ: X → X × X, x ↦ (x,x) is continuous 3. Tychonoff Theorem and Examples One of the most important theorems about product spaces is Tychonoff's Theorem , which extends compactness to arbitrary products. Torus: T² = S¹ × S¹ is compact (product of two compact spaces) n-Torus: Tⁿ = S¹ × ... × S¹ (n factors) is a compact manifold Cylinder: S¹ × [0,1] is compact Hilbert Cube: [0,1]^ℕ (infinite product) is compact by Tychonoff

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