Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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

share|improve this question

closed as not a real question by Asaf Karagila, Chris Eagle, Jennifer Dylan, William, J. M. Aug 21 '12 at 11:46

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

What is a weakly separable topological space? –  t.b. May 14 '12 at 2:46

1 Answer 1

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.

share|improve this answer

Not the answer you're looking for? Browse other questions tagged or ask your own question.