Closure of a set in a topological space
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Stub grade: A*
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There is an (ancient) page, Closure, interior and boundary, ensure all information there is present here and then turn that into an overview page. This page is marked A* in grade because of the importance of the closure, interior and boundary concepts
- Note: closure is an important term in mathematics (eg a group is "closed" under its operation), hence the specific name. This name must be inline with the closely related concepts of interior of a set in a topological space and boundary of a set in a topological space, boundary is the reason closure (topology) couldn't be used as even in topology "boundary" has several meanings.
- closure (set, topology) redirects here, for use with template:link.
Contents
Definition
Let [ilmath](X,\mathcal{ J })[/ilmath] be a topological space, let [ilmath]A\in\mathcal{P}(X)[/ilmath] be an arbitrary subset of [ilmath]X[/ilmath]. The closure of [ilmath]A[/ilmath], denoted [ilmath]\overline{A} [/ilmath], is defined as follows^{[1]}:
- [ilmath]\overline{A}:=\bigcap\{B\in\mathcal{P}(X)\ \vert\ A\subseteq B\wedge \underbrace{(X-B)\in\mathcal{J} }_{B\text{ is closed} } \}[/ilmath] - the intersection of all closed sets which contain [ilmath]A[/ilmath]
- Recall, by definition, that a set is closed if its complement is open and that [ilmath]X-B[/ilmath] is another way of writing the complement of [ilmath]B[/ilmath] in [ilmath]X[/ilmath]