Nonempty topologically convex space: Difference between revisions
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==Definition== | ==Definition== | ||
A '''topologically convex space''' is a [[topological space]] that is [[homeomorphism|homeomorphic]] to a [[convex subset of Euclidean space]]. | A '''nonempty topologically convex space''' is a nonempty [[topological space]] that is [[homeomorphism|homeomorphic]] to a [[convex subset of Euclidean space]]. | ||
==Relation with other properties== | ==Relation with other properties== | ||
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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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| [[Stronger than::topologically star-like space]] || || follows from [[convex implies star-like]] || || {{intermediate notions short|topologically star-like space|topologically convex space}} | | [[Stronger than::topologically star-like space]] || || follows from [[convex implies star-like]] || the [[pair of intersecting lines]] is topologically star-like but not topologically convex || {{intermediate notions short|topologically star-like space|topologically convex space}} | ||
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| [[Stronger than::contractible space]] || has a [[contracting homotopy]] || [[star-like implies contractible|via star-like]] || || {{intermediate notions short|contractible space|topologically convex space}} | | [[Stronger than::contractible space]] || has a [[contracting homotopy]] || [[star-like implies contractible|via star-like]] || || {{intermediate notions short|contractible space|topologically convex space}} | ||
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| [[Stronger than::semi-suddenly contractible space]] || has a [[semi-sudden homotopy|semi-sudden]] [[contracting homotopy]] | | [[Stronger than::semi-suddenly contractible space]] || has a [[semi-sudden homotopy|semi-sudden]] [[contracting homotopy]] || [[star-like implies contractible|via star-like]] || || {{intermediate notions short|semi-suddenly contractible space|topologically convex space}} | ||
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| [[Stronger than::SDR-contractible space]] || has a [[contracting homotopy]] that is a [[deformation retraction]] || [[star-like implies contractible|via star-like]] || || {{intermediate notions short|SDR-contractible space|topologically convex space}} | | [[Stronger than::SDR-contractible space]] || has a [[contracting homotopy]] that is a [[deformation retraction]] || [[star-like implies contractible|via star-like]] || || {{intermediate notions short|SDR-contractible space|topologically convex space}} | ||
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| [[Stronger than::equiconnected space]] || (complicated, described in one context as "contractible mod diagonal") || [[nonempty topologically convex implies equiconnected]] || || {{intermediate notions short|equiconnected space|nonempty topologically convex space}} | |||
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Latest revision as of 22:57, 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 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 |