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If X (topological space) is separable then X is weakly separable Thanks for your help

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What is a weakly separable topological space? –  t.b. May 14 '12 at 2:46
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closed as not a real question by Asaf Karagila, Chris Eagle, Jennifer Dylan, William, J. M. Aug 21 '12 at 11:46

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1 Answer

up vote 4 down vote accepted

If you’re talking about weak separability in the sense of Beshimov, meaning that $X$ has a $\sigma$-centred $\pi$-base, let $\tau$ be the topology on $X$, let $D=\{x_k:k\in\Bbb N\}$ be a countable dense subset of $X$, for each $k\in\Bbb N$ let $\mathscr{B}_k=\{B\in\tau:x_k\in B\}$, and let $\mathscr{B}=\bigcup\limits_{k\in\Bbb N}\mathscr{B}_k$; $\mathscr{B}$ is easily seen to be a $\sigma$-centred $\pi$-base for $\langle X,\tau\rangle$.

There is another notion of weak separability that applies only to metric spaces: $\langle X,d\rangle$ is weakly separable in this sense iff there is a sequence $\langle\varphi_k:k\in\omega\rangle$ of functions from $X$ to $\Bbb R$ such that $\operatorname{Lip}(\varphi_k)\le 1$ for all $k\in\Bbb N$ and $d(x,y)=\inf\limits_{k\in\Bbb N}|\varphi_k(x)-\varphi_k(y)|$ for all $x,y\in X$. If this is your definition, let $D$ be as above, and define $\varphi_k(x)=d(x,x_k)$ for each $x\in X$.

If yours is different from both of these, you’ll have to give us the definition.

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