Discovering Topological Invariants

Topology · Axiom Academy

INTRO Discovering Topological Invariants How do we prove two spaces are NOT homeomorphic? We've learned that two spaces are homeomorphic if there exists a continuous bijection with a continuous inverse between them. Homeomorphisms preserve topological structure - they're the "sameness" relation in topology. But here's the hard question: How do we prove that two spaces are NOT homeomorphic? We need a smarter approach. We need to find properties that homeomorphisms must preserve - properties so fundamental that if they differ between two spaces, we know immediately that no homeomorphism can exist. A topological invariant is a property of a space that is preserved under homeomorphism. If two spaces have different values for a topological invariant, they cannot be homeomorphic. Think of topological invariants as fingerprints of spaces. Just as no two people have the same fingerprints, if two spaces have different "topological fingerprints," they must be different (non-homeomorphic) spaces. Let's explore some of the most important topological invariants... Classic Topological Invariants Here are some of the most powerful and intuitive topological invariants: The number of "holes" in a surface Using Invariants to Distinguish Spaces Let's see how invariants solve our original problem. Consider these classical questions:

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