Defn: x ∈ X is a limit point of A iff x ∈ U^{open} implies U ∩ A {x} ≠ ∅.

Defn: A′ = the set of all limit points of A.

Thm 17.6: \overline{A} = A ∪ A′.

Cor 17.7: A closed if and only if A′ ⊂ A.

Defn: x_{n} converges to a limit x if for every neighborhood U of x, there exists a positive integer N such that n ≥ N implies x_{n} ∈ U.

Note: limit point of a set is not the same as limit of a sequence.

Defn: X is Hausdorff space if for all x_{1}, x_{2} ∈ X such that x_{1} ≠ x_{2}, there exists neighborhoods U_{1} and U_{2} of x_{1} and x_{2}, respectively, such that U_{1} ∩ U_{2} = ∅.

Thm 17.8: Every finite point set in a Hausdorff space X is closed.

Defn: X is T_{1} if every one point set is closed.

Thm 17.9: Let X by T_{1}, A ⊂ X. Then x is a limit point of A if and only if every neighborhood of x contains infinitely many points of A.

Thm 17.10: If X is Hausdorff, then a sequence of points of X converges to at most one point of X. (IFF?)

\frac{1}{n} converges to in the finite complement topology.

Thm 17.11: If X has the order topology, then X is Hausdorff. The product of two Hausdorff spaces is Hausdorff. A subspace of a Hausdorff space is Hausdorff.