Definition:Closure (Topology)/Definition 1
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Definition
Let $T = \struct {S, \tau}$ be a topological space.
Let $H \subseteq S$.
The closure of $H$ (in $T$) is defined as:
- $H^- := H \cup H'$
where $H'$ is the derived set of $H$.
That is, $H^-$ is the union of $H$ and its limit points.
Notation
The closure operator of $H$ is variously denoted:
- $\map \cl H$
- $\map {\mathrm {Cl} } H$
- $\overline H$
- $H^-$
Of these, it can be argued that $\overline H$ has more problems with ambiguity than the others, as it is also frequently used for the set complement.
$\map \cl H$ and $\map {\mathrm {Cl} } H$ are cumbersome, but they have the advantage of being clear.
$H^-$ is neat and compact, but has the disadvantage of being relatively obscure.
On $\mathsf{Pr} \infty \mathsf{fWiki}$, $H^-$ is notation of choice, although $\map \cl H$ can also be found in places.
Also see
- Results about set closures can be found here.
Sources
- 1953: Walter Rudin: Principles of Mathematical Analysis ... (previous) ... (next): $2.26$
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $3$: Continuity generalized: topological spaces: $3.7$: Definitions: Definition $3.7.11$
- 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
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Point Sets: $9.$
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): closure
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): closure
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): closure
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): accumulation point