Equivalence of Definitions of Weakly Locally Connected at Point/Definition 1 implies Definition 2

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.

$\blacksquare$