[Math] Prove that a continuous image of a closed subset of a compact space is a closed subset

compactnesscontinuitygeneral-topologymetric-spaces

Suppose $f$ is a continuous mapping from a compact metric space $X$ into a metric space $Y$. Prove that if $F$ is a closed subset of $X$, then $f[F]$ is a closed subset of $Y$.

Here is my idea for the proof: The continuous image of a connected space is connected. Use the intermediate value theorem to show that the image of every continuous real-valued function is an interval, and should return closed sets into closed sets.

Corrections are appreciated!

Best Answer

You might want to proceed in the following manner:

  1. Closed subsets of compact sets are compact, thus $F$ is compact;
  2. Continuous image of a compact set is compact, hence $f(F)$ is compact in $Y$;
  3. Compact sets in a metric space are closed, hence $f(F)$ is closed.
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