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Let $f: (M,d) \rightarrow (N,\rho)$ be continuous and let $A$ be a separable subset of M. Prove that $f(A)$ is separable.

My idea is to take countable dense subset of A and then construct an open ball for its points and use the definition of continuity in a metric space to conclude that $f(A)$ is separable.

Kindly help me to provide a rigorous proof.

Thanks!

Martin Sleziak
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gaufler
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  • @G.Chiusole When $A$ is said to be a separable subset. What does it mean ? 1. A is a Countable Dense subset in M. or 2. A has a countable dense subset. If it is the later than how should I proceed ? – gaufler Sep 15 '19 at 11:23
  • It means the latter. Proceed as is done in the question linked and the use the comments to the question (or the first answer). I the question I linked, $X$ is the separable subset ($A$ in your case) and $A$ is the countable dense subset – G. Chiusole Sep 15 '19 at 11:28

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