From continuity to homeomorphism, compactness in domain

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SUMMARY

The claim that all continuous bijections f: X → Y are homeomorphisms, given that all closed subsets of X are compact, is established as true under the assumption that Y is a Hausdorff space. The relevant theorem states that if X is compact and Y is Hausdorff, then any continuous bijection f: X → Y is indeed a homeomorphism. The proof relies on the fact that the image of a compact set under a continuous function is compact, and compact sets in Hausdorff spaces are closed. Thus, the equivalence of closed subsets being compact to X being compact is crucial for this conclusion.

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  • Understanding of topological spaces
  • Familiarity with the concepts of compactness and Hausdorff spaces
  • Knowledge of continuous functions and bijections in topology
  • Ability to follow mathematical proofs in topology
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  • Study the properties of compact spaces in topology
  • Learn about Hausdorff spaces and their significance in topology
  • Explore theorems related to continuous functions and homeomorphisms
  • Investigate the implications of compactness in various topological contexts
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Mathematicians, students of topology, and anyone interested in the foundational concepts of continuity and compactness in topological spaces.

jostpuur
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Is this claim true? Assume that X,Y are topological spaces, and that all closed subsets of X are compact. Then all continuous bijections f:X\to Y are homeomorphisms.

It looks true on my notebook, but I don't have a reference, and I don't trust my skills. Just checking.
 
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There is this useful theorem that says "If X is compact and Y is Hausdorff, then any continuous bijection f:X-->Y is a homeo" and the proof goes like this:

to show: f(C) is closed for all C in X closed. Take such a C. Since X is compact, C is compact, so f(C) is compact. But compact sets in a Hausdroff and closed, QED.

Observe that to say that all closed subsets of X are compact is equivalent to saying that X is compact. So, modulo Y being Hausdorff, your claim is the above theorem.
 
I see. I must have used the Hausdorff assumption without noticing it.
 

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