Which of the following are Hausdorff?

  • Thread starter sa1988
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In summary, the topologies {∅, {a,b}}, {∅, {a}, {a,b}}, and {∅, {b}, {a,b}} are not Hausdorff, whereas the topology {∅, {a}, {b}, {a,b}} is.
  • #1
sa1988
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Homework Statement



Which of the following topologies are Hausdorff (if any)?

{∅, {a,b} }

{∅, {a}, {a,b} }

{∅, {b}, {a,b} }

{∅, {a}, {b}, {a,b} }

Homework Equations



Definitions:

A neighbourhood U of x is an open set U⊂X such that xϵU

A topological space is Hausdorff if for each pair x, y of distinct points in X there exist neighbourhoods U of x and V of y which are disjoint.

The Attempt at a Solution

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Another description I've seen for the Hausdorff property is that for some two subsets U and V in X, the intersection of U and V is the empty set.

Looking at the given topologies, it seems that only the last one is Hausdorff. One can take {a} and {b} and see that their intersection is ∅. .

Hoping I'm correct with this? I just wanted to put it by the physics forum crowd because topology is a new and potentially worrying subject I've taken on at university, with a lot of new definitions and terminology being thrown around at quite a fast pace!

Thanks :oldsmile:
 
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  • #2
You are on the right track, but it might help to write out your reasoning. i.e. If you believe the first 3 are not Hausdorff, you should be able to say why, in symbols. You can use you "other description" to do so. i.e. Why (in symbols) do you believe the first isn't Hausdorff? Prove it!
 
  • #3
And yes topology definitions are going to come at you hard and fast.
braceyourself.jpg
 
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  • #4
dkotschessaa said:
You are on the right track, but it might help to write out your reasoning. i.e. If you believe the first 3 are not Hausdorff, you should be able to say why, in symbols. You can use you "other description" to do so. i.e. Why (in symbols) do you believe the first isn't Hausdorff? Prove it!

Alright, I'd say for the topologies ##\tau_n ## in the above question statement,
for ##n=[1,3] ## and taking any ##U, V \subset \tau## where ##U \not= V## and ##U, V \not= \emptyset##,
##U \cap V \not= \emptyset## for all ## U, V##, hence ##\tau_1, \tau_2, \tau_3## are not Hausorff

whereas for ##\tau_4## there exists ## U, V \subset \tau## such that ##U \cap V = \emptyset##, hence ##\tau_4## is Hausdorff.
 
  • #5
Try to prove that a finite space is Hausdorff if and only if it is discrete. That's a fun exercise.
 

Related to Which of the following are Hausdorff?

1. What is the definition of a Hausdorff space?

A Hausdorff space is a topological space in which any two distinct points have disjoint open neighborhoods. This means that for any two points in the space, there exists separate neighborhoods around each point that do not overlap.

2. How do you prove that a space is Hausdorff?

To prove that a space is Hausdorff, you must show that for any two distinct points in the space, there exists disjoint open neighborhoods around each point. This can be done by using the definition of a Hausdorff space and showing that the conditions are met.

3. What are some examples of Hausdorff spaces?

Examples of Hausdorff spaces include the real numbers with the standard topology, Euclidean spaces, and metric spaces. In general, any topological space that is T2 (Hausdorff) is also T1 (a space in which every singleton set is closed).

4. Can a non-Hausdorff space be made into a Hausdorff space?

Yes, it is possible to modify a non-Hausdorff space to make it into a Hausdorff space. This can be done by adding extra points or sets to the space, or by changing the topology in a way that satisfies the definition of a Hausdorff space.

5. Why is the concept of a Hausdorff space important in mathematics?

Hausdorff spaces are important in mathematics because they have many useful properties and allow for more precise and rigorous analysis of topological spaces. They also provide a foundation for many other important concepts in mathematics, such as compactness and continuity.

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