Equivalence of Definitions of Weakly Locally Connected at Point/Definition 1 implies Definition 2
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Theorem
Let $T = \struct {S, \tau}$ be a topological space.
Let $x \in S$.
Let $x$ have a neighborhood basis consisting of connected sets.
Then:
- Every open neighborhood $U$ of $x$ contains an open neighborhood $V$ of $x$ such that every two points of $V$ lie in some connected subset of $U$.
Proof
Let $U$ be an open neighborhood of $x$.
By assumption there exists a connected neighborhood $C$ of $x$ such that $C \subseteq U$.
By definition of a neighborhood, there exists an open neighborhood $V$ of $x$ such that $V \subseteq C$.
From Subset Relation is Transitive, $V \subseteq U$.
By definition of a subset:
- $\forall y, z \in V: y, z \in C$
The result follows.
$\blacksquare$