Regular or normal topological space

Question

How do we define an open neighborhood around a closed set? My question is with respect to a normal or regular topological space where we use the concept of an open neighborhood around a closed set.

Answer 1

Sorry, don't have enough points to comment; if someone could please turn this into a comment: maybe you can do this for each point x in the closed set C , take an open set $U_x$ containing x. Then $\cup U_x$ for all $x$ in $C$ is an open neighborhood containing $C$. For spaces like manifolds, you have , under the right conditions, regular neighborhoods or tubular neighborhoods.

Answer 2

Just like an open neighbourhood of $x$ is an open set that contains $x$, an open neighbourhood of a closed set $F$ is just an open set that contains it: $F \subseteq O$.

This is essentially the same as what @Lefschetz is saying in the other post: if we have $O$ we could pick $O$ for every point of $F$ as well, and if we pick an open neighbourhood for every point, their union is exactly an open superset of $F$.