Continuity and Homeomorphisms Summary

Topology · Axiom Academy

SUMMARY Unit 3: Continuity and Homeomorphisms Understanding when spaces are "the same shape" in topology In Unit 3, you've discovered the heart of topology: continuous functions and homeomorphisms. You've learned how to determine when a function preserves topological structure, when two spaces are topologically equivalent, and how to construct new spaces via quotient maps. These concepts unite everything you've learned so far and reveal what topology is really about. A function between topological spaces is continuous if the preimage of every open set in is open in : A function is a homeomorphism if: When a homeomorphism exists, we say and are homeomorphic , written . Given a space and an equivalence relation , the quotient space has the quotient topology: A set in is open if and only if its preimage under the quotient map is open in . 2. Key Takeaways: Continuous Functions Open set definition: Continuity is defined via preimages of open sets, not epsilon-delta (though they're equivalent for metric spaces) Multiple characterizations: A function is continuous iff preimages of open sets are open, iff preimages of closed sets are closed Composition preserves continuity: If and are continuous, then is continuous Local to global: Continuity can be checked on a basis — it's enough to verify preimages of basis elements are open Examples everywhere: Identity maps, constant maps, and inclusions are always continuous (with appropriate topologies)

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