# Category:Isolated Points

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This category contains results about Isolated Points in the context of Topology.
Definitions specific to this category can be found in Definitions/Isolated Points.

Let $T = \struct {S, \tau}$ be a topological space.

### Isolated Point of Subset

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

$\exists U \in \tau: U \cap H = \set x$

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 $T = \struct {S, \tau}$:

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

$\exists U \in \tau: U = \set x$

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

## Subcategories

This category has the following 3 subcategories, out of 3 total.

## Pages in category "Isolated Points"

The following 16 pages are in this category, out of 16 total.