Nonempty topologically convex space

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

A topologically convex space is a topological space that is homeomorphic to a convex subset of Euclidean space.

Relation with other properties

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
topologically star-like space		follows from convex implies star-like	the pair of intersecting lines is topologically star-like but not topologically convex	|FULL LIST, MORE INFO
contractible space	has a contracting homotopy	via star-like		Equiconnected space, Space in which every retraction is a deformation retraction, Topologically star-like space|FULL LIST, MORE INFO
semi-suddenly contractible space	has a semi-sudden contracting homotopy	via star-like		Topologically star-like space|FULL LIST, MORE INFO
SDR-contractible space	has a contracting homotopy that is a deformation retraction	via star-like		Space in which every retraction is a deformation retraction, Topologically star-like space|FULL LIST, MORE INFO