nLab
isolated point

Isolated points

Isolated points

Definitions

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

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

The antithetical concept is that of an accumulation point.

Properties

Every function on XX is continuous at PP if PP is isolated.

Created on June 28, 2020 at 05:04:41. See the history of this page for a list of all contributions to it.