Topology Closed Neighborhood Proof

The topology of $\mathbb{R}_U$ is the set $\{ \emptyset \} \cup \{ \mathbb{R} \} \cup \{ (r, \infty) | r \in \mathbb{R} \}$ .

The direction you did was the "hard" direction. I think you forgot that the whole space is always a closed set, since it's the complement of the empty set, which is always open.

Let $p \in \mathbb{R}$ . Then $p \in (p-1, \infty)$ , which is open. Let $N = \mathbb{R}$ . Will show $N$ is a closed neighborhood of $p$ .

Have that $N = \{\emptyset\}^c$ is closed. Since $p \in (p-1, \infty) \subset N$ , it follows that $N$ is a closed neighborhood of $p$ .