Theorem 3.15. Let (X,T) be a topological space, and let S be a collection of subsets of X. Then S is a subbasis for T if and only if (1) SC T , and (2) for each set U in T´ and point p in U there is a finite collection {V}*-1 of elements of S such that n PENKCU. i=1
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![**Theorem 3.15.** Let \((X, \mathscr{T})\) be a topological space, and let \(\mathcal{S}\) be a collection of subsets of \(X\). Then \(\mathcal{S}\) is a subbasis for \(\mathscr{T}\) if and only if
1. \(\mathcal{S} \subset \mathscr{T}\), and
2. for each set \(U\) in \(\mathscr{T}\) and point \(p\) in \(U\) there is a finite collection \(\{V_i\}_{i=1}^n\) of elements of \(\mathcal{S}\) such that
\[
p \in \bigcap_{i=1}^{n} V_i \subset U.
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