Linearly orderable space

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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
monotonically normal space |FULL LIST, MORE INFO
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
normal space T_1 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
completely regular space Monotonically normal space, Normal Hausdorff space|FULL LIST, MORE INFO
regular space Monotonically normal space, Normal Hausdorff space|FULL LIST, MORE INFO
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
T1 space points are closed linearly orderable implies T1, also via Hausdorff (via Hausdorff, others) Normal Hausdorff space|FULL LIST, MORE INFO