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
Weaker properties
| Property |
Meaning |
Proof of implication |
Proof of strictness (reverse implication failure) |
Intermediate notions
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| Hereditarily paracompact Hausdorff space |
Hausdorff space that is also a hereditarily paracompact space: every subspace (with the subspace topology) is a paracompact space |
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|FULL LIST, MORE INFO
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| Paracompact Hausdorff space |
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CW implies paracompact Hausdorff |
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|FULL LIST, MORE INFO
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| Perfectly normal space |
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CW implies perfectly normal |
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|FULL LIST, MORE INFO
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| Normal space |
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CW implies normal |
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Hereditarily normal space, Paracompact Hausdorff space, Perfectly normal space|FULL LIST, MORE INFO
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| Hausdorff space |
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CW implies Hausdorff |
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Completely regular space, Normal Hausdorff space, Regular Hausdorff space|FULL LIST, MORE INFO
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| Locally contractible space |
every point is contained in a contractible open subset |
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|FULL LIST, MORE INFO
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| Locally path-connected space |
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[[CW implies locally path-connected] |
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Locally contractible space|FULL LIST, MORE INFO
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| 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
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