When is an Open Set Homeomorphic to the Interior of its Closure?

Question

Let $X$ be a topological space and $U \subseteq X$ open. Then $U \subseteq \operatorname{int}(\operatorname{cl}(U))$.

I am looking for known assumptions on $X$ and $U$ such that one of the following properties holds:

1. $U$ and $\operatorname{int}(\operatorname{cl}(U))$ are homeomorphic
2. if $U$ and $\operatorname{int}(\operatorname{cl}(U))$ are homeomorphic then $U = \operatorname{int}(\operatorname{cl}(U))$, i.e. $U$ is a regular open set.

Examples and counterexamples:

1. If $X = \mathbb{R}$ then $U = (1,2) \cup (2,3)$ is not homeomorphic to $\operatorname{int}(\operatorname{cl}(U)) = (1,3)$.
2. If $X = \mathbb{R}^n$ and $U$ convex then $U$ is regularly open.

For my purposes, $X$ can be assumed to be locally compact and metrizable.