# 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 :)