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## Homework Statement

Prove that every Hausdorff topology on a finite set is discrete.

I'm trying to understand a proof of this, but it's throwing me off--here's why:

## Homework Equations

To be Hausdorff means for any two distinct points, there exists disjoint neighborhoods for those points.

Also, any finite subset of a Hausdorff space is closed.

## The Attempt at a Solution

Let a set X have n elements (I'll write it more formal later), but I'll denote them a 1,...,i,...,j,...n.

For each singleton element, we can write write it as:

{i} = [itex]\bigcap[/itex](X\{j}) s.t. j≠i.

And the set {i} is open because it's the intersection of open sets (X\{j}).

However, isn't that opposite of Hausdorff because both sets are finite subsets.

Thank you in advance for your help.