Thursday, December 29, 2011

Interesting Note on the Development of Set Theory

I was reading Walter Rudin's "Principles of Mathematical Analysis" and came across the notion of a perfect set.

"[A subset] E is perfect if E is closed and if every point of E is a limit point of E."

Of course, this also implies that, since a set in a topological space is closed if and only if it contains all of its limit points, E is exactly equivalent to the set of its limit points. I found this rather interesting, so I searched Wikipedia for an article on "perfect sets" and was redirected to the article on "Derived sets".

The article informed me that the derived set of a subset of a topological space is the set of its limit points (thus a perfect set is one equal to its own derived set). While I found this interesting, what I really found fascinating was a brief one-line note:

"The concept was first introduced by Georg Cantor in 1872 and he developed set theory in large part to study the derived sets on the real line."

I had always known that Cantor studied topology prior to developing set theory, but I had never been able to find the crucial link that led Cantor to develop the general theory of sets. Well, there you go. I love science history.