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 termed linearly orderable if it occurs as the underlying topological space of a linearly ordered space (viz it can be obtained by giving the order topology to a linearly ordered set).
Relation with other properties
Weaker properties
| Property |
Meaning |
Proof of implication |
Proof of strictness (reverse implication failure) |
Intermediate notions
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| monotonically normal space |
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|FULL LIST, MORE INFO
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| hereditarily normal space |
every subspace is a normal space |
(via monotonically normal) |
(via monotonically normal) |
Hereditarily collectionwise normal space, Monotonically normal space|FULL LIST, MORE INFO
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| normal space |
 and disjoint closed subsets can be separated by disjoint open subsets |
(via monotonically normal) |
(via monotonically normal) |
Collectionwise normal space, Hereditarily collectionwise normal space, Hereditarily normal space, Monotonically normal space|FULL LIST, MORE INFO
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| completely regular space |
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Monotonically normal space, Normal Hausdorff space|FULL LIST, MORE INFO
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| regular space |
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Monotonically normal space, Normal Hausdorff space|FULL LIST, MORE INFO
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| Hausdorff space |
distinct points can be separated by disjoint open subsets |
linearly orderable implies Hausdorff, also via others |
(via regular, normal) |
Monotonically normal space, Normal Hausdorff space|FULL LIST, MORE INFO
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| T1 space |
points are closed |
linearly orderable implies T1, also via Hausdorff |
(via Hausdorff, others) |
Normal Hausdorff space|FULL LIST, MORE INFO
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