Hausdorff topological spaces

In this video, we look at topological spaces which satisfy an extra "separation axiom". They are known as Hausdorff spaces, and they are interesting because the separation axiom ensures that limits, when they exist, are unique. We give examples of Hausdorff spaces, and also a non-Hausdorff space where limits are not unique. Finally, we talk about what it means for a property of spaces to be topological. We see that being Hausdorff is such a topological property.

[gargiborah]

Just awsome. Clear explanation . Please keep it up. Very helpful one.

[maraeacookson]

Awesome really enjoyed your vid, thank you from your neighbour in Gisborne NZ

[fbkintanar]

Thanks a lot for these videos, I am attending a course on general topology, and this has been helpful in developing conceptual understanding, which I don't get adequately from just doing a few exercises. For example, the distinction between open neighborhoods and open sets. Around 15:10, you draw a dashed line around a region in Y, and mention that it might be larger that the dashed region in domain Xaround the corresponding point x'. At first I thought, no that can't be right. If f is a homeomorphism, it must be a bijection on points, so it is also an injection on points. So the mapping from the the two regions must be something like an inclusion, you just renamed the points. Then I realized the the definition of Hausdorff is in terms of open *neighborhoods*, not open sets. And an open neighborhood is something that can be larger that the chosen open set. So although there is an injective map from the dashed region in X, seen as an open set and not just a neighborhood, to strictly smaller open set in Y, f could very well map to a larger open neighborhood that contains the smaller one, even if the larger neighborhood has more points. Now why does a larger neighborhood have to exist, and why does f have to be able to specify such a larger neighborhood? Let me think a bit more about that. I guess it doesn't have to exist for an arbitrary homeomorphism, there could some which will map to the smaller region and it will be maximal. But there could be other homeomorphisms which don't behave that way, and they would still carry over the Hausdorff condition. I think I need to learn more about separation axioms.
Thanks again for the help, and keep the videos coming. I'm already following the series on tensors.

[ragnaros]

This is the first video I am watching from your channel and I am truly impressed by your teaching skills! I wonder if you are planning on doing a video series on functional analysis at some point?

[parmeswaryadav]

Sir plz solve this qustion
Show that one point compactification of set of rational numbers Q is not Hausdorff.

[hyperduality]

Space is dual!
Distance, length, or space is defined by two distinct or separate dual points.
Up is dual to down, left is dual to right, in is dual to out -- space duality.
Points are dual to lines -- the principle of duality in geometry.
Space is dual to time -- Einstein.
Time is dual!
The future is dual to the past -- time duality.
We remember the past but predict the future.
Absolute time (Galileo) is dual to relative time (Einstein) -- time duality.
My absolute time is your relative time and your absolute time is my relative time -- time duality.
Space duality is dual to time duality.
"Always two there are" -- Yoda.