PlanetMath: T1 space

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T1 space

(Definition)

A topological space $(X,\tau)$ is said to be (or said to hold the axiom) if for all distinct points $x,y\in X$ ( $x\neq y$ ), there exists an open set $U\in\tau$ such that $x\in U$ and $y\notin U$ .

A space being is equivalent to the following statements:

For every $x\in X$ , the set $\{x\}$ is closed.
Every subset of is equal to the intersection of all the open sets that contain it.
Distinct points are separated.

"T1 space" is owned by drini. [ full author list (2) | owner history (1) ]

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Other names:

Attachments:: a space is T1 if and only if every singleton is closed (Proof) by waj
a space is T1 if and only if every subset A is the intersection of all open sets containing A (Proof) by waj
a space is if and only if distinct points are separated (Theorem) by matte

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Cross-references: separated, contain, intersection, subset, closed, equivalent, open set, points, axiom, topological space
There are 11 references to this entry.

This is version 7 of T1 space, born on 2002-02-08, modified 2005-05-18.
Object id is 1852, canonical name is T1Space.
Accessed 4836 times total.

Classification:

AMS MSC:

54D10 (General topology :: Fairly general properties :: Lower separation axioms )

Pending Errata and Addenda

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