Equivalence of Definitions of Local Basis/Local Basis for Open Sets Implies Neighborhood Basis of Open Sets
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Theorem
Let $T = \struct {S, \tau}$ be a topological space.
Let $x$ be an element of $S$.
Let $\BB$ be a set of open neighborhoods of $x$ such that:
- $\forall U \in \tau: x \in U \implies \exists H \in \BB: H \subseteq U$
Then $\BB$ satisfies:
- every neighborhood of $x$ contains a set in $\BB$.
Proof
Let $N$ be a neighborhood of $x$.
Then there exists $U \in \tau$ such that $x \in U$ and $U \subseteq N$ by definition.
By assumption, there exists $H \in \BB$ such that $H \subseteq U$.
From Subset Relation is Transitive, $H \subseteq N$.
The result follows.
$\blacksquare$