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.

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.

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 U_i avoids y_i. The intersection of U with all the U_i  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.