Topology Lecture 07: Hausdorff Spaces

We define what it means to be a Hausdorff space. Then we show that in Hausdorff spaces finite sets are closed, limits of convergent sequences are unique, and neighborhoods of limit points intersect the limiting set in infinitely many points.

00:00 Introduction
00:19 Motivation for Hausdorff Property
08:58 Definition: Hausdorff Space
09:35 Example: Metric Spaces
13:00 Example: Discrete Spaces
14:18 Prop: Finite subsets of Hausdorff spaces are closed
19:42 Prop: Limits are unique in Hausdorff spaces
24:50 Prop: Neighborhoods of limit points contain infinitely many points of limiting set

This lecture follows Lee's "Introduction to topological manifolds", chapter 2.

A playlist with all the videos in this series can be found here:

[darrenpeck]

I used to think group theory lectures were badly done but actually topology is much, much harder and usually the lectures are not good. These are fantastic!

[darrenpeck]

Beautiful lecture, using Hausdorff property. Thank you. I hope you make more playlists!!!

[yang]

Very good lecture on point set topology! Very clear and helpful to give motivations of definitions!

[leandrocarg]

Also the only convergent sequences in a discrete space are the eventually constant sequences. That's why these spaces are not interesting from the point of view of analysis, as well as non Hausdorff spaces.
Great videos, keep it up. I hope your channel grows up!

[matheusjahnke]

18:32 "why is this?"... because picking any point p ∈ X \ {f0}, it is proven that p is in the interior of X \ {f0} (as in, there's a neighborhood of p completely contained in X \ {f0})... which means the interior of X \ {f0} is equal to X \ {f0}... which means X \ {f0} is open;

[ChrisRossaroDidatticaDigitale]

16:27 I reckon you meant one point Hausdorff space is closed.

[sgut]

Hi Marius. Thanks for these very interesting videos.
I think there's a problem with the last proposition: let X={a, b, c, d} and suppose that T is the power set of X, i.e. every subset of X is open. Then T is the discrete topology on X. So (X, T) is Hausdorff, by your second example. Alternatively place a, b, c and d at the corners of a square and use Manhattan distance as a metric. Then we have a metric space, Hausdorff by your first example.
But X is finite, so how can any limit point contain infinitely many points of any set?
In the illustration of the last proposition, you have placed p outside A. It's a big assumption that this is possible in finite spaces.
Am I missing something?

[shengzheyang]

I'm confused about the beginning example in lecture 7 that 2 is also a limit point of set {2, 2, 2, 2...}. According to the definition of limit point in lecture 3, a point p is a limit point of a set if any open set containing p contains points of the set other than the point p itself. There are two open sets that contain 2, which are {1, 2, 3} and {1, 2}, and neither them meets the requirement of the definition, then how could 2 be a limit point of the constant set {2, 2, 2, 2, ...}?

[mollicoHD]

In the last proposition, why does a1 necessarily have to lie in U_p? Why does the new the neighborhood of p disjoint from a0 have to intersect A inside of U?

[awazin]

I really love your playslist :) I have a question which torments me in the example with the constant sequence 2. If you add the open space (1, 3), 3 is still in the neighborhood of 2? What I did not understand ? I thank you for your help :)

[jtinmoscow]

For the last proposition, how do we know/prove that these sets are infinite?

[AgolaOdero]

Is there such a thing as a limit of a set? Isn't it always a limit of a sequence?

[AlessandroOrjuela]

For the last property, what if you consider a topology where individual points are open? Eg. The discrete topology. Then won’t the property not hold?

Second example—what if your topological space is finite? Eg. What if it’s {1, 2, 3}? I imagine you can still have a Hausdorff space on that—in which case this would be contradicted—but maybe you can’t