Definition:Weakly Locally Connected at Point/Definition 1

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 $x$ has a neighborhood basis consisting of connected sets.

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

Sources

• 2000: James R. Munkres: Topology (2nd ed.): $3$: Connectedness and Compactness: $\S 25$: Components and Local Connectedness: Exercise $6$