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It is not true in general that in a topological space, the closure of the interior of a set $A$ is the closure of $A$. This seems to be true, however, if $A$ is a convex subset of $\mathbb{R}^n$ in the usual topology. This property seems to assert that $A$ is somewhat "well-behaved", but what is the name of this property?

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If you take the segment $[(0,0),(0,1)]\subset \mathbb{R}^2$, it's interior is empty and it's closure is not empty, but this segment is convex. –  Tomás Jan 30 '13 at 13:50
See here for the correct statement. This post may also be of interest. –  David Mitra Jan 30 '13 at 13:52
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In the special case when $A$ is closed, it is called a regular closed set. This article discusses the dual notion of a regular open set; the properties listed there are easy to dualize. The word "regular" is overused, of course.

I do not know any established term for general sets with this property. I'd simply say "a set with dense interior".

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