Prove every Hausdorff topology on a finite set is discret.

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  • #1
Hodgey8806
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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.
 

Answers and Replies

  • #2
Dick
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{i} is closed, since it's a finite subset of a Hausdorff space. Now you've proved {i} is also open. So it's clopen. Doesn't that make it a discrete space?
 
  • #3
Hodgey8806
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Ah ok!

My book mentions that every subset of a discrete space is closed, but it doesn't explictly say that it is open when we first discussed them. It mentioned the topology is that every set is open...so I suppose it's implied.

Thank you!
 
  • #4
Dick
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Ah ok!

My book mentions that every subset of a discrete space is closed, but it doesn't explictly say that it is open when we first discussed them. It mentioned the topology is that every set is open...so I suppose it's implied.

Thank you!

Right. If every set S is closed then its complement is also closed. So S is also open.
 

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