# Category:Boundaries

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This category contains results about Boundaries in the context of Topology.

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

Let $H \subseteq S$.

### Definition from Closure and Interior

The **boundary of $H$** consists of all the points in the closure of $H$ which are not in the interior of $H$.

Thus, the **boundary of $H$** is defined as:

- $\partial H := H^- \setminus H^\circ$

where $H^-$ denotes the closure and $H^\circ$ the interior of $H$.

## Subcategories

This category has only the following subcategory.

### B

## Pages in category "Boundaries"

The following 21 pages are in this category, out of 21 total.

### B

- Boundary is Intersection of Closure with Closure of Complement
- Boundary of Boundary is Contained in Boundary
- Boundary of Boundary is not necessarily Equal to Boundary
- Boundary of Boundary of Subset of Indiscrete Space
- Boundary of Compact Closed Set is Compact
- Boundary of Compact Set in Hausdorff Space is Compact
- Boundary of Empty Set is Empty
- Boundary of Intersection is Subset of Union of Boundaries
- Boundary of Set is Closed
- Boundary of Subset of Discrete Space is Null
- Boundary of Subset of Indiscrete Space
- Boundary of Union is Subset of Union of Boundaries
- Boundary of Union of Separated Sets equals Union of Boundaries