Difference between revisions of "Closure of a set in a topological space"

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Latest revision as of 10:16, 28 September 2016

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.

Definition

Let (X,J) be a topological space, let AP(X) be an arbitrary subset of X. The closure of A, denoted ¯A, is defined as follows[1]:

  • ¯A:={BP(X) | AB(XB)JB is closed} - the intersection of all closed sets which contain A
    • Recall, by definition, that a set is closed if its complement is open and that XB is another way of writing the complement of B in X

See also

References

  1. Jump up Introduction to Topological Manifolds - John M. Lee