Let $f:(X, \tau_x) \to (Y,\tau_y)$ be a continuous and onto function. I need to show that if $X$ is separable (Lindelöf), then $Y$ is respectively separable (Lindelöf).
Transferability of space properties via continuous functions
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general-topology
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0@ Brian you are right, I just checked my book it sounded good to assume equality:), sorry.Thank you so much! – 2012-11-16
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You need to assume in addition that $f$ maps $X$ onto $Y$.
(1) Let $D$ be a countable dense subset of $X$, and consider the set $f[D]$ in $Y$. It’s certainly countable; can you show that it’s dense?
(2) Let $\mathscr{U}$ be an open cover of $Y$, and let $\mathscr{V}=\{f^{-1}[U]:U\in\mathscr{U}\}$; use the continuity of $f$ to conclude that $\mathscr{V}$ is an open cover of $X$. Let $\mathscr{W}$ be a countable subcover of $\mathscr{V}$; can you see where to go from here to finish up?
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1@Klara: Or for each $V\in\mathscr{V}$ you could pick a $U_V\in\mathscr{U}$ such that $V=f^{-1}[U_V]$ and let $\mathscr{G}=\{U_V:V\in\mathscr{V}\}$, but since $f$ is onto, it comes to the same thing and does indeed give you the desired countable subcover. – 2012-11-16