1, closing point
Let s be a subset of Euclidean space. If all tee shots centered on X contain a point in S (this point can also be X itself), then X is called the closed point of S. ..
The above definition can be extended to any subset S of metric space X. Specifically, let X be a metric space, where the metric is d and S is a subset within X. If for all r >；; 0, there is a point y in s, which makes d (x, y)
In addition, it can be defined as follows: if d(x, S):=inf{d(x, s):sinS}=0, and x is called the closure point of S, the above two definitions are written in the same way.
Finally, the definition of closure point can also be extended to topological space, just replace "kick-off" with neighborhood. Let s be a subset of topological space X, then X is called a closed point of S. If all X neighborhoods contain a point in S, please note that this definition does not require neighborhoods to be open sets.
2. Limit point
The definition of closed point is very close to the definition of limit point. The difference between these two definitions is small but important-in the definition of limit point, the neighborhood of point X must contain points in this set that are "not X's own".
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.
3. Closure of a set
The closure of the set S refers to the set consisting of all closure points of S, and the closure of S is written as cl(S), Cl(S) or S? .