Closure (topology)

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In mathematics, the closure of a subset S in a topological space consists of all points in S plus the limit points of S. Intuitively, these are all the points that are "near" S. A point which is in the closure of S is a point of closure of S. The notion of closure is in many ways dual to the notion of interior.

Contents

Definitions

Point of closure

For S a subset of a Euclidean space, x is a point of closure of S if every open ball centered at x contains a point of S (this point may be x itself).

This definition generalises to any subset S of a metric space X. Fully expressed, for X a metric space with metric d, x is a point of closure of S if for every r > 0, there is a y in S such that the distance d(x, y) < r. (Again, we may have x = y.) Another way to express this is to say that x is a point of closure of S if the distance d(x, S) := inf{d(x, s) : s in S} = 0.

This definition generalises to topological spaces by replacing "open ball" or "ball" with "neighbourhood". Let S be a subset of a topological space X. Then x is a point of closure of S if every neighbourhood of x contains a point of S. Note that this definition does not depend upon whether neighbourhoods are required to be open.

Limit point

The definition of a point of closure is closely related to the definition of a limit point. The difference between the two definitions is subtle but important — namely, in the definition of limit point, every neighborhood of the point x in question must contain a point of the set other than x itself.

Thus, every limit point is a point of closure, but not every point of closure is a limit point. A point of closure which is not a limit point is an isolated point. In other words, a point x is an isolated point of S if it is an element of S and if there is a neighbourhood of x which contains no other points of S other than x itself.

For a given set S and point x, x is a point of closure of S if and only if x is an element of S or x is a limit point of S (or both).

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