Geometric Intuition of regular open sets In Oxtoby's Measure and Category book, he has the following definition at page $20.$

A regular open set is a set that is equal to the interior of its closure. 

If I am not mistaken, by definition, $A = int (\overline{A})$ if and only if $A$ is regular open.

Question: What is its geometric intuition of the definition?

It seems that open set may not be regular open, for example, $A=(1,2)\cup(2,3).$
Also, closed set is not regular open, as we have $int(\overline{A}) = int({A})$ is open and $A$ is closed. 

The geometric intuition is that it has no “skinny holes” (holes without interior).
The closure fills the isolated holes, and the interior then makes it open again.
Note that for general sets, the operation also removes "skinny islands" (skinny holes of the complement; removed during forming the interior), but open sets don't have them to begin with.
Also note that the operation maps an arbitrary set to a regular open set.
