Prove every Hausdorff topology on a finite set is discret.

In summary, to prove that every Hausdorff topology on a finite set is discrete, we can show that every singleton element is both open and closed, making it a discrete space. This is because in a Hausdorff space, every finite subset is closed, and in a discrete space, every subset is both open and closed.
  • #1
Hodgey8806
145
3

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.
 
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  • #2
{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
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
Hodgey8806 said:
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.
 

Related to Prove every Hausdorff topology on a finite set is discret.

1. What is the definition of a Hausdorff topology?

A Hausdorff topology is a type of topological space in which any two distinct points have disjoint neighborhoods. This means that for every pair of points in the space, there exists open sets that contain one point but not the other.

2. Why is it important to prove that every Hausdorff topology on a finite set is discret?

This proof is important because it helps us understand the structure of finite topological spaces. It also shows that every finite topological space has a very simple structure, making it easier to analyze and classify.

3. What is a discret topology?

A discret topology is a type of topological space in which every subset is open. This means that the only open sets in a discret topology are the empty set and the entire space. In other words, every point in the space is an isolated point.

4. How can we prove that every Hausdorff topology on a finite set is discret?

To prove this, we can use the fact that any finite set has a finite number of subsets. Since a Hausdorff topology requires that every pair of distinct points have disjoint neighborhoods, we can construct a neighborhood for each point that contains only that point. This way, we can show that every subset of the finite set is open, and therefore the topology is discret.

5. Can this proof be extended to infinite sets?

No, this proof only applies to finite sets. In infinite sets, there are infinite pairs of distinct points, making it impossible to construct neighborhoods that contain only one point. Therefore, not all Hausdorff topologies on infinite sets are discret.

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