If a set is separable then it has countable bases. But what about converse??. If a set has countable bases is it separable. Any help would be appreciated. Thanks.
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1Possible duplicate of Second Countable, First Countable, and Separable Spaces – martin.koeberl Mar 29 '17 at 15:24
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What makes you think that if a set is separable it has a countable base? – bof Mar 30 '17 at 04:55
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A topological space (not a set) is called separable iff it has a countable dense subset. Countable means not-uncountable, that is, finite or countably infinite.
A topological space is called second-countable iff it has a countable base. A space can be separable without being second-countable. There are many examples.
If a space is second-countable then it is separable. Because if $B$ is a countable base, then for each non-empty $b\in B$ choose $f(b)\in b$. Every non-empty open set $S$ has some non-empty $b\in B$ as a subset, so $S$ has $f(b)$ as a member. So the countable set $D=\{f(b): \phi\ne b\in B\}$ intersects every non-empty open set, so $D$ is dense.
There are many other interesting cardinals associated with a topology. They are referred to as topological cardinal functions.
DanielWainfleet
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