I am really short of time, so let me summarize. I have joined the Institute of Mathematical Sciences for my PhD. Right now, I am desperately trying to avoid being succumbed to the pressure of grueling coursework here. But a mathematician is a masochist, so I am enjoying the mathematical agony!

A famous theorem in point-set topology due to Kuratowski was stated as an assignment problem in our topology class. With a hint from our Topology professor, I solved the problem which I regard as the best problem in point-set topology I ever solved yet. The problem seems  too fascinating to be even true. It appeared ﬁrst in a paper due to Kuratowski and was made popular by Kelley in his book General Topology and can be found as a “starred” problem in Munkres’ Topology. I could solve it, not mainly because of the hint  given by my professor but with the comfort that someone has already solved the problem. It does seem daunting to begin working on an open problem. Without further ado, I state the problem:

Let $(X, \tau)$ be a topological space and $A\subseteq X$. By iteratively applying operations of closure and complemention, one cannot produce more than 14 disjoint sets.

The proof of this theorem is found in the adjoining section “Notes” or by clicking here.