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Two questions: Is the property of being separable topological, and is the continuous image of a separable metric space separable? I have intuition that the answer to the first question is true and the answer to the second question is false, but am confused as to how to begin the proofs. Any guidance or hints would be appreciated. Thanks!

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Both are true. If $X$ is separable, with countable dense subset $D$, and $f:X\to Y$ is a continuous surjection, then $f[D]$ is a countable dense subset of $Y$. Let $U$ be any non-empty open set in $Y$; then $f^{-1}[U]$ is a non-empty open set in $X$, so there is an $x\in f^{-1}[U]\cap D$, and clearly $f(x)\in U\cap f[D]$.

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  • $\begingroup$ U be any nonempty set in Y.....think there is a typo... $\endgroup$ – user8795 Oct 2 '16 at 9:20
  • $\begingroup$ @user8795: There is no typo. $\endgroup$ – Brian M. Scott Oct 2 '16 at 18:53

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