# Category:Definitions/Isolated Points

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This category contains definitions related to Isolated Points in the context of Topology.

Related results can be found in Category: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 $x$.

## Pages in category "Definitions/Isolated Points"

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

### I

- Definition:Isolated Point
- Definition:Isolated Point (Topology)
- Definition:Isolated Point (Topology)/Space
- Definition:Isolated Point (Topology)/Subset
- Definition:Isolated Point (Topology)/Subset/Definition 1
- Definition:Isolated Point (Topology)/Subset/Definition 2
- Definition:Isolated Point of Subset
- Definition:Isolated Point of Topological Space