nLab isolated point

Definitions

A point $P$ of a topological space $X$ is isolated if it is a neighbourhood of itself, in other words if the singleton subset $\{P\}$ is open.

More generally, a point $P$ of a subset $A$ of a space $X$ is isolated in $A$ if it is isolated when viewed as a point in the subspace $A$ with the subspace topology. More explicitly, for some neighbourhood $U$ of $P$ (in $X$), $U \cap A = \{P\}$.

The antithetical concept is that of an accumulation point.

Properties

Every function on $X$ is continuous at $P$ if $P$ is isolated.