# Definition:Weakly Locally Connected at Point

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## 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**.