= Solution
The image $f(K)$ is <compact>: pull any open cover back by the <continuous map>, use a finite subcover of $K$, then map forward. For a general topological target, closedness is not guaranteed; a <continuous> image into the indiscrete two-point space can be a nonclosed singleton. No <connectedness> assertion follows merely from <compactness> of $K$.
If $Y$ is metric, $f(K)$ is <compact>, therefore closed in $Y$, <bounded> and <totally bounded>. It is also <complete> in its induced metric and <sequentially compact>. For the latter properties, <compact> <metric spaces> have a convergent subsequence of every sequence; a <Cauchy sequence> with one convergent subsequence converges to that same limit. Total <boundedness> follows by selecting finitely many balls of each prescribed radius. These are properties of the image itself, irrespective of whether $Y$ is <complete>.
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