- #1

jostpuur

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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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- Thread starter jostpuur
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In summary, the conversation discusses whether a claim is true or not. The claim states that if X and Y are topological spaces and all closed subsets of X are compact, then all continuous bijections from X to Y are homeomorphisms. The speaker mentions a useful theorem that supports this claim and provides a proof. It is mentioned that X being compact and Y being Hausdorff is necessary for the claim to hold.

- #1

jostpuur

- 2,116

- 19

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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- #2

quasar987

Science Advisor

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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.

- #3

jostpuur

- 2,116

- 19

I see. I must have used the Hausdorff assumption without noticing it.

Continuity is a mathematical concept that describes the smoothness and connectedness of a function. A function is continuous if small changes in the input result in small changes in the output.

Homeomorphism is a type of continuous function that is also bijective, meaning it has a one-to-one correspondence between its inputs and outputs. This means that a homeomorphism preserves the continuity of the function.

Compactness in domain refers to a mathematical property of a set, where every open cover of the set has a finite subcover. In simpler terms, this means that the set does not have any "gaps" or "holes" and can be covered by a finite number of smaller sets.

Compactness in domain is important in mathematics because it allows for easier analysis and manipulation of functions. It also has many applications in other fields, such as physics and economics.

Compactness in domain is often used in real-world problems to find optimal solutions or to determine the existence of solutions. For example, in economics, compactness in domain can be used to prove the existence of an equilibrium point in a market system.

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