If $X$ is a compact metric space and $f:X \to Y$ is a continuous map , where $Y$ is another metric space , then is $f(X)$ a complete subset of $Y$ ?
2026-04-03 08:50:48.1775206248
If $X$ is a compact metric space and $f:X \to Y$ is a continuous map , where $Y$ is another metric space , then is $f(X)$ a complete subset of $Y$ ?
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If $X$ is compact and $f$ is continuous, then $f(X)$ is a compact. Moreover since $Y$ is a metric space, $f(X)\subset Y$ is also a metric space. Finally, you know that every metric space that are compact are complete, therefore $f(X)$ is complete.