Product Space Examples

Algebraic Topology · Axiom Academy

EXAMPLE Product Space Constructions Building new topological spaces through Cartesian products Excellent work! You've mastered product space constructions. Here's what we learned: Product topology: The product topology on X × Y is generated by basis elements U × V where U is open in X and V is open in Y, making projections continuous Universal property: A map f: Z → X × Y is continuous if and only if both component maps π X ∘ f and π Y ∘ f are continuous Cylinder construction: S 1 × [0,1] gives the cylinder, showing how products naturally encode geometric structure Torus from circles: T 2 = S 1 × S 1 provides an alternative to the quotient construction, revealing deeper structure Property preservation: Products preserve compactness (Tychonoff), connectedness, and Hausdorff properties from factor spaces Higher dimensions: The n-torus T n = (S 1 ) n generalizes naturally, providing fundamental examples in higher-dimensional topology Product spaces and quotient spaces are dual constructions, both essential for building and analyzing topological spaces in algebraic topology.

This is the written version of the interactive lesson above. See the full Algebraic Topology course.