Nonempty topologically convex 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
A nonempty topologically convex space is a nonempty 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 | |
| equiconnected space | (complicated, described in one context as "contractible mod diagonal") | nonempty topologically convex implies equiconnected | |FULL LIST, MORE INFO |