Closure of a set in a topological space

From Maths
(Redirected from Closure (topology))
Jump to: navigation, search
Stub grade: A*
This page is a stub
This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:
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.


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]

See also


  1. Introduction to Topological Manifolds - John M. Lee