CW-space

From Topospaces
Jump to: navigation, search
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 topological space is said to be a CW-space if it possesses a CW-decomposition, or in other words, if it can be viewed as the underlying topological space of a CW-complex.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Polyhedron occurs as the geometric realization of a simplicial complex |FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Hereditarily paracompact Hausdorff space Hausdorff space that is also a hereditarily paracompact space: every subspace (with the subspace topology) is a paracompact space |FULL LIST, MORE INFO
Paracompact Hausdorff space CW implies paracompact Hausdorff |FULL LIST, MORE INFO
Perfectly normal space CW implies perfectly normal |FULL LIST, MORE INFO
Normal space CW implies normal Hereditarily normal space, Paracompact Hausdorff space, Perfectly normal space|FULL LIST, MORE INFO
Hausdorff space CW implies Hausdorff Completely regular space, Normal Hausdorff space, Regular Hausdorff space|FULL LIST, MORE INFO
Locally contractible space every point is contained in a contractible open subset |FULL LIST, MORE INFO
Locally path-connected space [[CW implies locally path-connected] Locally contractible space|FULL LIST, MORE INFO
Homotopy-CW-space homotopy-equivalent to a CW-space (obvious) any contractible space that is not a Hausdorff space, e.g., the line with two origins |FULL LIST, MORE INFO