For a topological space $(X,T)$ with a basis $B$, is every basis element of $B$ an open set of $X$ (i.e. in $T$)?
(Forgive me for the dumb question, just trying to learn the basics)
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By definition $B$ is a base for $T$ if and only if $T$ is the set of all unions of subsets of $B$. For any $b\in B$, $\{b\}$ is a subset of $B$, so its union is in $T$. But its union is just the set $b$, so $b\in T$. Thus, $B\subseteq T$. |
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(just so the question does not remain unanswered) By definition, any arbitrary union has to be in $T$ hence also a `union' with one element. |
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