Category:Set Boundaries
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This category contains results about Set Boundaries in the context of Topology.
Definitions specific to this category can be found in Definitions/Set Boundaries.
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 the following 2 subcategories, out of 2 total.
B
E
- Examples of Set Boundaries (6 P)
Pages in category "Set Boundaries"
The following 23 pages are in this category, out of 23 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 Cartesian Product of Subsets
- 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