a space is T1 if and only if every singleton is closed

Say X is a http://planetmath.org/node/1852T1 topological spaceMathworldPlanetmath. Let’s show that {x} is closed for every xX:

The T1 axiom (http://planetmath.org/T1Space) gives us, for every y distinct from x, an open Uy that contains y but not x. Since we’re in a topological space, we can take the union of all these open sets to get a new open set,


{x} is the complement of U, closed because U is open: None of the Uy contain x, so U doesn’t contain x. But any yx is in U, since yUyU. That takes care of that.

Now let’s say we have a topological space X in which {x} is closed for every xX. We’d like to show that T1 holds:

Given xy, we want to find an open set that contains x but not y. {y} is closed by hypothesisMathworldPlanetmath, so its complement is open, and our search is over.

Title a space is T1 if and only if every singleton is closed
Canonical name ASpaceIsT1IfAndOnlyIfEverySingletonIsClosed
Date of creation 2013-03-22 14:20:15
Last modified on 2013-03-22 14:20:15
Owner waj (4416)
Last modified by waj (4416)
Numerical id 7
Author waj (4416)
Entry type Proof
Classification msc 54D10
Related topic ASpaceIsT1IfAndOnlyIfEverySubsetAIsTheIntersectionOfAllOpenSetsContainingA