# Topologically star-like 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

A non-empty topological space is termed **topologically star-like** if it is homeomorphic to a star-like subset of Euclidean space (Where the Euclidean space is possibly infinite-dimensional), where the latter is endowed with the subspace topology from Euclidean space.

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

topologically convex space | homemorphic to a convex subset of Euclidean space | follows from convex implies star-like | ? | |FULL LIST, MORE INFO |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions | |
---|---|---|---|---|---|

contractible space | has a contracting homotopy | star-like implies contractible | |FULL LIST, MORE INFO | ||

semi-suddenly contractible space | has a semi-sudden contracting homotopy | star-like implies contractible | |FULL LIST, MORE INFO | ||

SDR-contractible space | has a contracting homotopy that is a deformation retraction | star-like implies contractible | |FULL LIST, MORE INFO |