Definition:Weakly Locally Connected at Point

Definition

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

Let $x \in S$.

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 $U$.

Also known as

If $T$ is weakly locally connected at $x$, it is also said to be connected im kleinen at $x$.

Some sources refer to a space which is weakly locally connected at $x$ as locally connected at $x$.

Also see

• Equivalence of Definitions of Weakly Locally Connected at Point
• Equivalence of Definitions of Locally Connected Space where it is shown that a space that is weakly locally connected at every point is locally connected.
• Definition:Locally Connected Space
• Definition:Locally Path-Connected Space

Linguistic Note

The phrase im kleinen is German and means on a small scale.