Limit point
In mathematics, the limit point is a topological concept that profitably generalizes the notion of limit. The notion of a limit point is the conceptual underpinning of concepts such as closed set and topological closure. Indeed, one can categorize a closed set as a set that contains all of
its limit points. The topological closure operation can be defined as an operation that enriches a set by adding the limit points.
Proof. Let S be a closed set and x&isinX
a limit point thereof. Then, x must be in S, for otherwise
the complement of S would constitute an open neighborhood of x that does not intersect S.
Conversely, suppose that S contains all of its limit points.
We shall show that the complement of S is an open set. Let x¬inS be given. By assumption x is not
a limit point, and hence there exists an open neighborhood U of x that contains only finitely many points of S, call them
<math>y_1,\ldots y_n</math>
Using
the T0 assumption, we may choose open neighborhoods
<math>U_1,\ldots, U_n</math>
of x, such that Ui avoids yi. The intersection of U with all the
Ui produces an open neighborhood of x that
avoids S altogether. This proves that the complement of
S is open, and therefore that S is closed.
Q.E.D.
Formal Treatment Definition.
Let X be a topological space and S &sube X a subset thereof.
We say that a point x &isin X is a limit point (alternatively: accumulation point, cluster point) of S if
every open set containg x also contains infinitely many points of S.
Proposition 1.
Suppose that X is a T0 space (see Kolmogorov space, Separation axiom). A set S&subeX is closed if and only if it contains all
of its limit points.