Topologically star-like space: Difference between revisions
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! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | ! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | ||
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| [[Weaker than::topologically convex space]] || homemorphic to a [[convex subset of Euclidean space]] || follows from [[convex implies star-like]] || ? || {{intermediate notions short|topologically star-like space|topologically convex space}} | | [[Weaker than::nonempty topologically convex space]] || homemorphic to a [[convex subset of Euclidean space]] || follows from [[convex implies star-like]] || ? || {{intermediate notions short|topologically star-like space|topologically convex space}} | ||
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Revision as of 22:17, 26 October 2023
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 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 |
|---|---|---|---|---|
| nonempty 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 |