Topology Problem With Continuous Function and Ultrafilter

Question

Let X and Y be compact topological spaces. Prove that a function f:X-->Y is continuous if and only if for every for every family (xi) in X and every ultrafilter U on I, the following holds: f(lim xi) =lim f(xi) over I and U.

Solution Preview

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We first try to prove the equivalence: A space is compact if and only if every ultrafilter on it is convergent.

Proof: =>
We assume first that X is compact and U being an ultrafilter on it.
We let C= { A ̅ | A ∈ U} //accumulation points
Now, all finite subfamilies of C have a nonempty intersection and because X...

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