# Definition. Let (X, J) be a topological space, A a subset of X, and p a point in X. Then p is a limit point of A if and only if for each open set U containing p, (U -{p}) nA # 0.

Definition. Let (X, J) be a topological space, A a subset of X, and p a point in X. Then p
is a limit point of A if and only if for each open set U containing p, (U -{p}) nA # 0.
Notice that p may or may not belong to A.
Definition. Let (X, J) be a topological space, A a subset of X, and p a point in X. If
p E A but p is not a limit point of A, then p is an isolated point of A.
Definition. Let (X, J) be a topological space and A C X. Then the closure of A in X,
denoted A or CI(A) or Clx(A), is the set A together with all its limit points in X.
Definition. Let (X, J) be a topological space and A C X. The subset A is closed if and
only if A = A, in other words, if A contains all its limit points.
Theorem (1) For any topological space (X, J) and A C X, the set A is closed. That is,
for any set A in a topological space, A = A.
(2) Let (X, J) be a topological space. Then the set A is closed if and only if
X – A is open.

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The post Definition. Let (X, J) be a topological space, A a subset of X, and p a point in X. Then p is a limit point of A if and only if for each open set U containing p, (U -{p}) nA # 0. appeared first on Superb Professors.