Definition:Weakly Locally Connected at Point/Definition 2
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Definition
Let $T = \struct {S, \tau}$ be a topological space.
Let $x \in S$.
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
Sources
- 1970: Stephen Willard: General Topology: Chapter $8$: Connectedness: $\S27$: Pathwise and local connectedness: Definition $27.14$
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): locally connected
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): locally connected