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Separated sets

In topology and related branches of mathematics, separated sets are pairs of subsets of a given topological space that are related to each other in a certain way: roughly speaking, neither overlapping nor touching. The notion of when two sets are separated or not is important both to the notion of connected spaces (and their connected components) as well…

Definitions, Relation to separation axioms and separated spaces & Relation to topologically distinguishable points

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Definitions

DisjointIntersectionEmpty setSet theoryClosure
19 related topics

Relation to separation axioms and separated spaces

DistinctHausdorff spaces
2 related topics

Relation to topologically distinguishable points

Open setSingleton sets
2 related topics

Relation to connected spaces

Complement
1 related topics

Key facts & relationships

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completely T2
(completely Hausdorff)
T0
(Kolmogorov)
T1
(Fréchet)
T2
(Hausdorff)
T2½
(Urysohn)
T3
(regular Hausdorff)

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Overview

Definitions

Relation to separation axioms and separated spaces

Relation to connected spaces

Relation to topologically distinguishable points

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Map overview Semantic statistics

Separated sets

Nodes38
Edges37
Triples16
Avg. degree1.95
Density0.052632
Components1

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Separated sets

Top relations

related to Relation to separation axioms and separated spaces · 6
Separated sets → As, Hausdorff, Separated, Specifically, T2, The
completely T2 · 1
Separated sets → (completely Hausdorff)
T0 · 1
Separated sets → (Kolmogorov)
T1 · 1
Separated sets → (Fréchet)
T2 · 1
Separated sets → (Hausdorff)
T2½ · 1
Separated sets → (Urysohn)
T3 · 1
Separated sets → (regular Hausdorff)
T3½ · 1
Separated sets → (Tychonoff)
T4 · 1
Separated sets → (normal Hausdorff)
T5 · 1
Separated sets → (completely normal Hausdorff)

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Important terminology

separated sets displaystyle topological two disjoint neighbourhoods space spaces closed separation set mathbb function given connected axioms open example point

Entity relationships Subject–Predicate–Object triples

SubjectPredicateObjectConfidenceSrc
Separated setscompletely T2(completely Hausdorff)1.00infobox
Separated setsT0(Kolmogorov)1.00infobox
Separated setsT1(Fréchet)1.00infobox
Separated setsT2(Hausdorff)1.00infobox
Separated setsT2½(Urysohn)1.00infobox
Separated setsT3(regular Hausdorff)1.00infobox
Separated setsT3½(Tychonoff)1.00infobox
Separated setsT4(normal Hausdorff)1.00infobox
Separated setsT5(completely normal Hausdorff)1.00infobox
Separated setsT6(perfectly normal Hausdorff)1.00infobox
Separated setsrelated to Relation to separation axioms and separated spacesThe0.60section
Separated setsrelated to Relation to separation axioms and separated spacesAs0.60section

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