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Canonical static concept record

$T_1$ space

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Summary

A topological space in which singletons are closed.

Record metadata

Canonical name: $T_1$ space
Concept ID: t1_space
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Topology
Domains: Topology
Alternate terminology: No alternate terminology is currently authored for this record.

Carrier(s)

topological space $X$

Data

existing topology

Axioms / constraints

$T_{1}$ separation axiom

Concept sources

Wikipedia — Separation axiom — Separation axiom · expository · source ID wp-separation
Morris — Topology Without Tears — Topology Without Tears · textbook · source ID topology-without-tears

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

$T_0$ space $T_1$ space

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: e_t0_t1
Relation type: Impose axiom impose-axiom
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: + $T_1$

Authored explanation

Require singletons to be closed.

How to interpret this relation type

Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.

Relation sources

Wikipedia — Separation axiom — Separation axiom · expository · source ID wp-separation
Morris — Topology Without Tears — Topology Without Tears · textbook · source ID topology-without-tears

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.

$T_1$ space Hausdorff space

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This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: e_t1_haus
Relation type: Impose axiom impose-axiom
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: + disjoint neighborhoods

Authored explanation

Require distinct points to have disjoint neighborhoods.

How to interpret this relation type

Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.

Relation sources

Wikipedia — Separation axiom — Separation axiom · expository · source ID wp-separation
Morris — Topology Without Tears — Topology Without Tears · textbook · source ID topology-without-tears

$T_1$ space Normal $T_1$ space

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: e_t1_normal
Relation type: Impose axiom impose-axiom
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: + normal separation

Authored explanation

Separate disjoint closed sets by neighborhoods.

How to interpret this relation type

Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.

Relation sources

Wikipedia — Normal space — Normal space · expository · source ID wp-normal-space
Morris — Topology Without Tears — Topology Without Tears · textbook · source ID topology-without-tears

$T_1$ space Regular $T_1$ space

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: e_t1_regular
Relation type: Impose axiom impose-axiom
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: + regular separation

Authored explanation

Separate points from closed sets by neighborhoods.

How to interpret this relation type

Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.

Relation sources

Wikipedia — Regular space — Regular space · expository · source ID wp-regular-space
Morris — Topology Without Tears — Topology Without Tears · textbook · source ID topology-without-tears

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$T_1$ space

Summary

Record metadata

Carrier(s)

Data

Axioms / constraints

Concept sources

Incoming relations (arrows to this concept)

$T_0$ space $T_1$ space

Authored explanation

How to interpret this relation type

Relation sources

Outgoing relations (arrows from this concept)

$T_1$ space Hausdorff space

Authored explanation

How to interpret this relation type

Relation sources

$T_1$ space Normal $T_1$ space

Authored explanation

How to interpret this relation type

Relation sources

$T_1$ space Regular $T_1$ space

Authored explanation

How to interpret this relation type

Relation sources