Homeomorphisms Review

Algebraic Topology · Axiom Academy

Understanding topological equivalence through bicontinuous bijections 1. Definition of Homeomorphism A homeomorphism is a continuous bijection with a continuous inverse. It establishes a "topological equivalence" between spaces. A function f: X → Y is a homeomorphism if: 1. f is bijective (one-to-one and onto) We write X ≅ Y and say "X is homeomorphic to Y" The key insight: being a continuous bijection is NOT enough! The inverse must also be continuous. A continuous bijection need not be a homeomorphism! The inverse might fail to be continuous. Let X = [0, 2π) with the subspace topology from ℝ Let Y = S¹ (unit circle) with the standard topology Define f: X → Y by f(t) = (cos t, sin t) Then: f is a continuous bijection, but f⁻¹ is NOT continuous! Problem: [0, π) is open in X, but f([0, π)) is not open in S¹ A topological invariant is a property preserved by homeomorphisms. These help us prove spaces are NOT homeomorphic. Important Topological Invariants: • Connectedness (a connected space can't be homeomorphic to a disconnected one) • Compactness (compact ≠ non-compact) • Hausdorff property (separation axioms) • Fundamental group (from algebraic topology!) • Homology groups (also from algebraic topology) 4. Classic Examples and Non-Examples Building intuition through concrete examples is crucial for understanding homeomorphisms. • (0, 1) ≅ ℝ (open interval ≅ entire real line) • Any open interval ≅ any other open interval • A square ≅ a circle ≅ a triangle (all in ℝ²)

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