closed point

Let X be a topological spaceMathworldPlanetmath and suppose that xX. If {x}={x}¯ then we say that x is a closed point. In other words, x is closed if {x} is a closed set.

For example, the real line equipped with the usual metric topologyMathworldPlanetmath, every point is a closed point.

More generally, if a topological space is T1 (, then every point in it is closed. If we remove the condition of being T1, then the property fails, as in the case of the Sierpinski space X={x,y}, whose open sets are , X, and {x}. The closurePlanetmathPlanetmath of {x} is X, not {x}.

Title closed point
Canonical name ClosedPoint
Date of creation 2013-03-22 16:22:24
Last modified on 2013-03-22 16:22:24
Owner jocaps (12118)
Last modified by jocaps (12118)
Numerical id 9
Author jocaps (12118)
Entry type Definition
Classification msc 54A05