closed point

Let $X$ be a topological space and suppose that $x\in X$ . If $\{x\}=\overline{\{x\}}$ 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 $\mathbb{R}$ equipped with the usual metric topology, every point is a closed point.

More generally, if a topological space is $T_{1}$ (http://planetmath.org/T1), then every point in it is closed. If we remove the condition of being $T_{1}$ , then the property fails, as in the case of the Sierpinski space $X=\{x,y\}$ , whose open sets are $\varnothing$ , $X$ , and $\{x\}$ . The closure 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