Equivalence of Definitions of Weakly Locally Connected at Point

Theorem

Let $T = \struct {S, \tau}$ be a topological space.

Let $x \in S$.

The following definitions of the concept of Weakly Locally Connected at Point are equivalent:

Definition 1

The space $T$ is weakly locally connected at $x$ if and only if $x$ has a neighborhood basis consisting of connected sets.

Definition 2

The space $T$ is weakly locally connected at $x$ if and only if 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

Definition 1 implies Definition 2

Let $x$ have a neighborhood basis consisting of connected sets.

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.

$\Box$

Definition 2 implies Definition 1

Let every open neighborhood $U$ of $x$ contain an open neighborhood $V$ of $x$ such that every two points of $V$ lie in some connected subset of $U$.

Let $\BB = \set {B \subseteq S: B \text{ is a connected neighborhood of } x}$

It will be shown that $\BB$ is a neighborhood basis consisting of connected sets.

Let $N$ be any neighborhood of $x$.

By definition of a neighborhood there exists an open neighborhood $U$ of $x$ such that $U \subseteq N$

By assumption there exists an open neighborhood $V$ of $x$ such that every two points of $V$ lie in some connected subset of $U$.

For each $y \in V, y \ne x$, let $C_y$ be a connected subset of $U$ such that $x, y \in C_y \subseteq U$.

Let $C = \set x \cup \bigcup \set {C_y:y \in V, y \ne x}$

From Singleton is Connected in Topological Space, $\set x$ is connected.

From Union of Connected Sets with Common Point is Connected, $C$ is a connected subset.

From Union is Smallest Superset, $C \subseteq U \subseteq N$.

Now $x \in C$ and for each $y \in V, y \ne x$ then $y \in C_y \subseteq C$.

Hence $V \subseteq C$.

By definition of a neighborhood, $C$ is a connected neighborhood of $x$.

That is, $C \in \BB$.

Since $N$ was an arbitrary neighborhood of $x$, it follows that $\BB$ is a neighborhood basis of $x$ consisting of connected sets.

$\blacksquare$