# Locally connected space

From Topospaces

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

## Contents

## Definition

### Equivalent definitions in tabular format

No. | Shorthand | A topological space is termed locally connected if ... |
---|---|---|

1 | locally connected at every point | for every point , and every open subset of containing , there exists an open subset of such that , , and is a connected space with the subspace topology. |

2 | weakly locally connected at every point | for every point , and every open subset of containing , there exists a subset of such that is in the interior of , , and is a connected space with the subspace topology. |

3 | basis of open connected subsets | 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

- Connected space: Being connected does not imply being locally connected, and being locally connected does not imply being connected.
`Further information: connected not implies locally connected, locally connected not implies connected`

### 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 |