# Category:Definitions/Isolated Points

This category contains definitions related to Isolated Points in the context of Topology.
Related results can be found in Category:Isolated Points.

Let $T = \left({S, \tau}\right)$ be a topological space.

### Isolated Point of Subset

$x \in H$ is an isolated point of $H$ if and only if:

$\exists U \in \tau: U \cap H = \left\{{x}\right\}$

That is, if and only if there exists an open set of $T$ containing no points of $H$ other than $x$.

### Isolated Point of Space

When $H = S$ the definition applies to the entire topological space $T = \left({S, \tau}\right)$:

$x \in S$ is an isolated point of $T$ if and only if:

$\exists U \in \tau: U = \left\{{x}\right\}$

That is, if and only if there exists an open set of $T$ containing no points of $S$ other than $x$.

## Pages in category "Definitions/Isolated Points"

The following 8 pages are in this category, out of 8 total.