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I'm taking a course in analysis and I was thinking about the relation between compact sets and homeomorphism. We know that if f is an onto and one-to-one homeomorphism then it follows that for every subset K:

K is compact in M <=> f(K) is compact in N

Now, does this go the other way too? That is, given that for every subset K that if K is compact then f(K) is compact, and vice versa, does it follow that f is an homeomorpism?

I've been trying to prove this by contradiction by assuming that f isn't continuous, taking a convergent series in K that isn't convergent in f(K) and trying to show somehow that this leads to that f(K) isn't compact, but so far no luck.

Any ideas on how to proceed, or if this in fact true?