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pre image of a closed set is closed

Source: Iran PPCE 2012-Analysis exam- P1

February 14, 2012
functionreal analysisreal analysis unsolved

Problem Statement

Suppose that XX and YY are two metric spaces and f:X⟶Yf:X \longrightarrow Y is a continious function. Also for every compact set K⊆YK \subseteq Y, it's pre-image fpre(K)f^{pre}(K) is a compact set in XX. Prove that ff is a closed function, i.e for every close set C⊆XC\subseteq X, it's image f(C)f(C) is a closed subset of YY.
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