TITLE: How to prove that $ \text{int}(\text{cl}(A)) = \text{cl}(\text{int}(A)) $?
QUESTION [14 upvotes]: How can one prove that $ \text{int}(\text{cl}(A)) = \text{cl}(\text{int}(A)) $, where $ A \subseteq \mathbb{R}^{n} $?
This is true for all sets that I have tried, but I can’t prove it formally.

REPLY [6 votes]: The following interactive page lets you vary the initial set $ E $ to see (among other things) whether or not it satisfies the identity $ \text{int}(\text{cl}(E)) = \text{cl}(\text{int}(E)) $:
http://mathdl.maa.org/images/upload_library/60/bowron/k14.html
For more on this topic and others like it, this page has plenty of references:
http://www.mathtransit.com/cornucopia.php