Hausdorff topology on five-element set that is not the discrete top.

1. Jan 29, 2013

ArcanaNoir

1. The problem statement, all variables and given/known data

The textbook exercise asks for a Hausdorff topology on $\{a,b,c,d,e\}$ which is not the discrete topology (the power set). It is from "Introduction to Topology, Pure and Applied", by Adams and Franzosa.

2. Relevant equations

Let X be a set.

Definition of topology (top for short): must include X and the empty set, must include all intersections of finitely many sets in the top, must include all unions of any sets in the top.

Definition of Hausdorff: for any two elements x and y of X, there must be disjoint open sets in the top such that one contains x and the other contains y.

3. The attempt at a solution

My thought is that this cannot be done. We have a theorem that if X is Hausdorff, then every single point subset of X is closed. This implies that all the four-element subsets of X are (in the topology) and open. But then the intersection of any of the four-element sets generates the rest of the powerset.

2. Jan 29, 2013

micromass

Staff Emeritus
That is correct. Any finite Hausdorff space must be discrete!

Your proof is correct as well!

3. Jan 29, 2013

ArcanaNoir

Thanks micro :)