Locally connected space

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This article defines a property of topological spaces: a property that can be evaluated to true/false for any topological space|View a complete list of properties of topological spaces

Definition

Equivalent definitions in tabular format

No. Shorthand A topological space X is termed locally connected if ...
1 locally connected at every point for every point x \in X, and every open subset V of X containing x, there exists an open subset U of X such that x \in U, U \subseteq V, and U is a connected space with the subspace topology.
2 weakly locally connected at every point for every point x \in X, and every open subset V of X containing x, there exists a subset A of X such that x is in the interior of A, A \subseteq V, and A is a connected space with the subspace topology.
3 basis of open connected subsets X has a basis (of open subsets) such that all members of the basis are connected in the subspace topology.

Relation with other properties

Incomparable properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
locally path-connected space
locally simply connected space
locally contractible space
locally Euclidean space
manifold

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
space in which all connected components are open
space in which the connected components coincide with the quasicomponents