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.