Intersection of a closure

Question

What is the intersection of a closure?

That is, $U$ is a open set, then what is:

Answer 1

The notation $$ \bigcap_{U\text{ open}, x\in U}\bar U $$ means: Intersect the closures of all open sets $U$ which contain $x$. The point $x$ is fixed, and $U$ is variable. Alternatively, you could use the notation $$ \bigcap \{\bar U \mid U\text{ open}, x\in U\} $$ In a Hausdorff space, this intersection is equal to $\{x\}$, and this equality (for all $x\in X$) is also sufficient for $X$ to be a Hausdorff space.