# 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 ClosedPoint 2013-03-22 16:22:24 2013-03-22 16:22:24 jocaps (12118) jocaps (12118) 9 jocaps (12118) Definition msc 54A05