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.