Closing point

For the subset S of Euclidean space, X is the closed point of S, if all kicks centered on X contain the point of S (this point can also be X).

This definition can be extended to any subset S of metric space X. Specifically, for metric space X with metric D, X is the closed point of S, if for all R >；; The existence of 0 and y belongs to S, so the distance d (x, y).

This definition can also be extended to topological space, but the "kick-off" is replaced by neighborhood. Let s be a subset of topological space X, then X is the closure point of S, if all X neighborhoods contain S points. Note that this definition does not require the neighborhood to be open.

limit point

The definition of closed point is very close to the definition of limit point. The difference between these two definitions is very small, but it is very important-in the definition of limit point, the neighborhood of point X must contain points with different sets from X.

So all limit points are closed points, but not all closed points are limit points. Either the closed point of the limit point or the isolated point. In other words, point X is an isolated point. If it is an element of S and has a neighborhood of X, then there is no other point in this neighborhood except X.

For a given set S and point X, X is the closed point of S if and only if X belongs to S, or X is the limit point of S.

Closure of a set

The closure of the set S is a set composed of all closure points of S. The closure of S is written as cl(S), Cl(S) or S? . The closure of a set has the following properties:

Cl(S) is the closed parent set of S.

Cl(S) is the intersection of all closed sets containing S.

Cl(S) is the smallest closed set containing s.

Set s is a closed set if and only if S = cl(S).

If s is a subset of t, then cl(S) is a subset of cl(T).

If A is a closed set, then A contains S if and only if A contains cl(S).

Sometimes the second or third property mentioned above is used as the definition of topological closure.

In the first countable space (such as metric space), cl(S) is all the limits of the convergent sequence of all points.

Note that the above properties are still valid if the words such as closure, intersection, inclusion, minimal and closure are replaced by internal, union, full, maximum and open. For more information, see "Close Operation" below.

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