Convergence

Suppose that \( \left( \mathcal{ T }, X \right) \) is a topological space, then a sequence of points \( \left( x_n \right) \) in \( X \) is said to converge to the point \( c \in X \) provided that for each neighborhood of \( c \), there exists some \( N \in \mathbb { N }_1 \) such that for all \( n \ge N \), we have \( x_n \in U \)

Hausdorff Space

A topological space \( X \) is called a Hausdorff space if for each pair of points \( x_1, x_2 \) of distinct points of \( X \), there exist neighborhoods \( U_1, U_2 \) of \( x_1, x_2 \) respectively that are disjoint

Every one point set is Closed in a Hausdorff Space

Suppose that \( p \in X \) where \( X \) is a Hausdorff space, then \( \left\{ p \right\} \) is closed

We will show that \( \left\{ p \right\} \) is closed by showing \( \left\{ p \right\} = \overline{ \left\{ p \right\} } \)

Suppose that \( y \in X \) and \( y \neq p \), since \( X \) was Hausdorff, we get \( U_p, U_y \) disjoint neighborhoods. Since \( U_y \cap U_p = \emptyset \) , since \( U_p \supseteq \left\{ p \right\} \), then \( U_y \cap \left\{ p \right\} = \emptyset \) therefore \( y \notin \overline{ \left\{ p \right\} } \).

Since \( \left\{ p \right\} \subseteq \overline{ \left\{ p \right\} } \), then we know \( p \in \overline{ \left\{ p \right\} } \), so therefore \( \overline{ \left\{ p \right\} } = \left\{ p \right\} \) as needed.

Every Finite Set is Closed in a Hausdorff Space