Definition:Boundary (Topology)/Definition 1
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Definition
Let $T = \struct {S, \tau}$ be a topological space.
Let $H \subseteq S$.
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 $H$.
Also known as
The boundary of a subset $H$ of a topological space $T$ is also seen referred to as the frontier of $H$.
Notation
The boundary of $H$ is variously denoted (with or without the brackets):
- $\partial H$
- $\map {\mathrm b} H$
- $\map {\mathrm {Bd} } H$
- $\map {\mathrm {fr} } H$ or $\map {\mathrm {Fr} } H$ (where $\mathrm {fr}$ stands for frontier)
- $H^b$
The notations of choice on $\mathsf{Pr} \infty \mathsf{fWiki}$ are $\partial H$ and $H^b$.
Also see
- Results about set boundaries can be found here.
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $1$: General Introduction: Closures and Interiors