Stop! This post contains spoilers. This page has the solution to the problem posed in yesterday’s post.

We challenged you to find a set from which we can make as many new sets as possible using only the closure and complement operations.

In 1922 Kuratowski proved the following theorem.

**Theorem**. At most 14 sets can be obtained from (including itself) using the closure and complement operations.

This is often called Kuratowski’s closure-complement theorem or Kuratowski’s 14-set theorem. (Note: he proved this for general topological spaces, not just .)

In fact, we can achieve this value. An example of such a set is…

We leave it to the reader to verify that we can produce 14 sets from using closures and complements.

James Fife gave a nice, readable proof (subscription required) of this theorem in 1991 (*Mathematics Magazine*, Vol. 64, No. 3 (Jun., 1991), pp. 180-182).

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*Related*

An interactive solution to the Kuratowski Closure-Complement Theorem can be found at

http://www.kuratowski.com

The interactive page that used to be at kuratowski.com is now at:

http://mathdl.maa.org/mathDL/60/?pa=content&sa=viewDocument&nodeId=3343

Thanks for the update!

That’s a great applet. Thank you for sharing it.