What is the Significance of Completely Regular Spaces?

[micromass]

Other than that, I don't find it useful to answer by basically throwing the book at me... telling me I should go and start from scratch.

Everybody here on this forum learned topology this way. It really is the best (and I guess the only) way to learn topology.

Instead, it might make it easier to answer if you could cut and pasted from some on-line source. Then I could read it and the context it's in and decide for myself if I need to go back to the very beginning.

Nobody learned topology from an online source. At least: nobody who has a good grasp on the material. If you want to get good in mathematics, then you have to start reading textbooks and doing exercises. It's the only way. The faster you start, the better.

 

[WannabeNewton]

The presentations and books I've seen don't really show how the concepts connect to the functions one would see in physics.

Just to say one last thing, why in the world would they? They are pure math books so of course there won't be "applications" to physics. The books are meant for you to actually learn the subject properly. If all you want a summary of topology and how it's used in physics but don't properly want to have a deep knowledge of the subject, appreciate its many counter examples and intricacies and elegance, then there are always mathematical physics books like Nakahara.

 

[friend]

Just to say one last thing, why in the world would they? They are pure math books so of course there won't be "applications" to physics. The books are meant for you to actually learn the subject properly. If all you want a summary of topology and how it's used in physics but don't properly want to have a deep knowledge of the subject, appreciate its many counter examples and intricacies and elegance, then there are always mathematical physics books like Nakahara.

"deeper knowledge of the subject"... I should "read a book"... I think this is a cop-out. All I asked is about the definition of completely regular spaces in terms of topologies, open and closed sets, and continuity. These are basic concepts, and I should think that you should be able to easily explain how all these things fit in the definition. But all I'm getting is excuses. The definition is only one paragraph long, and you're not even quoting part of it in any explanation you offer. At this point I'm not confident that you know what you're talking about. If it were so clear to you, you should be able to just cut and paste from some book you like. We don't seem to be communication very well. Maybe you should let someone else reply if they wish.

 

[WannabeNewton]

Yeah good luck with that. I should have read from wikipedia instead of using an actual textbook, then I could at least say I know what I'm talking about. Cheers!

 

[micromass]

Yeah, you're on to us! We don't actually know what we're talking about. I'm sorry for wasting your time. I'll let people reply who know topology now.

 

[friend]

Yeah, you're on to us! We don't actually know what we're talking about. I'm sorry for wasting your time. I'll let people reply who know topology now.

Or at least someone who will not refer me to obscurity. If it's so easy, quote the page. That's better than telling me to go read a book.

 

[jgens]

Maybe you should let someone else reply if they wish.

Others will likely be less patient with you than micromass and WannabeNewton, and despite what you seem to think, they really do know what they are talking about. Instead of complaining, consider taking their advice.

 

[strangerep]

Friend,

Since (apparently) you decline to participate in the quiz I posed in post #64 (which was designed to reveal and correct a basic point that you misunderstand), I'll just mention 2 more books before quitting this thread:

1) Many years ago, I was perplexed about all this "topology" stuff that the experts on sci.physics.research often talked about. Then someone mentioned this book:

Albert Schwarz, "Topology for Physicists",
https://www.amazon.com/dp/3642081312/?tag=pfamazon01-20

Very expensive, but I lashed out and bought a copy. But I couldn't get much out of it, and it still sits gathering dust on my shelf to this day.

2) Semour Lipschutz, Schaum Outline of General Topology,
https://www.amazon.com/dp/0071763473/?tag=pfamazon01-20

It's not a book aimed at physicists, but it is very concise and cheap, with lots of worked examples. When I started reading, I couldn't put it down. Sad, perhaps, but at least that shows it was exactly what I needed, even though it wasn't aimed at physicists.

--------

And BTW, you're deeply wrong about Micromass and WannabeNewton. I'm reminded of a story about when Einstein was asked by a reporter "can you explain your theory of relativity to me", he answered simply "No." When she asked why not, he said "could you explain to someone how to bake a cake if they don't know what flour is?". She was quite taken aback by this, and (probably) offended. But does this mean Einstein didn't understand relativity? Of course not.

 

[friend]

And BTW, you're deeply wrong about Micromass and WannabeNewton. I'm reminded of a story about when Einstein was asked by a reporter "can you explain your theory of relativity to me", he answered simply "No." When she asked why not, he said "could you explain to someone how to bake a cake if they don't know what flour is?". She was quite taken aback by this, and (probably) offended. But does this mean Einstein didn't understand relativity? Of course not.

As I've been told: you don't really understand anything unless you can explain it to someone else.

 

[micromass]

Well, this thread has run its course. So I'm locking it.

If anybody is interested in helping the OP further by answering his questions, then send me a PM and I will open the thread again.

Edit: Thread opened on request of dx.

 

[dx]

friend, here's an intuitive way to think about what a topology is and what an open set is, that might help understanding why you need a topology to speak about open sets:

A 'topology' on a space gives us a way to talk about points "sufficiently close" to members of that space, and the concept of open set can be described as "a set that contains all points sufficiently close to its members". Thus you need a topology to talk about open sets.

For example, the idea of an 'isolated point' can be defined in the language of open sets as a point x such that the set which contains only x is open, i.e. "if there are no points sufficiently close to the point"

 

[friend]

friend, here's an intuitive way to think about what a topology is and what an open set is, that might help understanding why you need a topology to speak about open sets:

A 'topology' on a space gives us a way to talk about points "sufficiently close" to members of that space, and the concept of open set can be described as "a set that contains all points sufficiently close to its members". Thus you need a topology to talk about open sets.

For example, the idea of an 'isolated point' can be defined in the language of open sets as a point x such that the set which contains only x is open, i.e. "if there are no points sufficiently close to the point"

Thank you, dx. But I already accept the need for open sets and unions and intersections thereof to describe topology. And I also accept that open sets cannot be described without topology. For at least any open set and the empty set also describe a topology.

I started this thread to understand the definition of completely regular spaces. The parts I don't get is where the closed set F comes from and if it's unique or arbitrary or constructed from the complement of open sets already in the topology. I'm also not clear about how the function f:X→[0,1] can be continuous when f is either 0 or 1 even up to the sharp boundary of the closed set F. I was hoping these things were easy to answer, but I may have been mistaken.

 

[dx]

The parts I don't get is where the closed set F comes from and if it's unique or arbitrary or constructed from the complement of open sets already in the topology.

A set is a closed if its complement is open, so the open sets determine what the closed sets are. And its not unique. We are not talking about some particular closed set F. For any closed set F, and any point x outside it, there must be a continuous function that separates them.

I'm also not clear about how the function f:X→[0,1] can be continuous when f is either 0 or 1 even up to the sharp boundary of the closed set F.

The definition only says that f must be 0 at x, and 1 on F. It says nothing about what its value must be anywhere else, except that the function must be continuous.

 

[Bacle2]

A set is a closed if its complement is open, so the open sets determine what the closed sets are. And its not unique. We are not talking about some particular closed set F. For any closed set F, and any point x outside it, there must be a continuous function that separates them.



The definition only says that f must be 0 at x, and 1 on F. It says nothing about what its value must be anywhere else, except that the function must be continuous.

But f does not have to be either 0 or 1. And remember that in abstract topological spaces (mostly not R^n and not manifolds) , the concept of borders does not really apply.

 

[dx]

I didn't say it had to be either 0 or 1. I said it has to be 0 on x, and 1 on F. Outside that, it can be anything it wants, as long its continuous.

And friend used the word 'boundary', not border, and boundary is a notion that applies in any topological space.

 

[Bacle2]

I didn't say it had to be either 0 or 1. I said it has to be 0 on x, and 1 on F. Outside that, it can be anything it wants, as long its continuous.

And friend used the word 'boundary', not border, and boundary is a notion that applies in any topological space.

Sorry, I was addressing a coment made by friend, not to your post; in the last paragraph of the most recent post by friend:

"I started...... I'm also not clear about how the function f:X→[0,1] can be continuous when f is either 0 or 1 even up to the sharp boundary of the closed set F. I was hoping these things were easy to answer, but I may have been mistaken. "

 

[dx]

Oh, sorry :) You quoted my post so I assumed you were replying to me.

 

[Bacle2]

Sorry myself for mistakenly quoting your post.

 

[friend]

A set is a closed if its complement is open, so the open sets determine what the closed sets are. And its not unique. We are not talking about some particular closed set F. For any closed set F, and any point x outside it, there must be a continuous function that separates them.

I may have questions about this later.

But first,

The definition only says that f must be 0 at x, and 1 on F. It says nothing about what its value must be anywhere else, except that the function must be continuous.

Yea, I've looked at 3 or 4 definitions on the Web. And they all seem a bit terse. So I'm having trouble visuallizing what the definition means. Does [0,1] mean that f must be somewhere in the closed interval 0≤f≤1?

Am I right in this interpretation:

You have a closed set, F, inside an entire set. (Nevermind for the moment where F comes from.) Chose a point x outside F and chose a point y inside F. Then there must be a continuous function f such that f(x)=0, and f(y)=1. The function f is defined for all points in the entire topology with no discontinuities. Is this right?

 

[dx]

Does [0,1] mean that f must be somewhere in the closed interval 0≤f≤1?

Yes.

Am I right in this interpretation:

You have a closed set, F, inside an entire set. (Nevermind for the moment where F comes from.) Chose a point x outside F and chose a point y inside F. Then there must be a continuous function f such that f(x)=0, and f(y)=1. The function f is defined for all points in the entire topology with no discontinuities. Is this right?

Yes, but you don't choose a particular y in F. The continuous function must be 1 for all y in F.

 

[friend]

Yes, but you don't choose a particular y in F. The continuous function must be 1 for all y in F.

It also seems in the definition you have "for any point x not in F", f(x)=0. So it seems that no matter what point x in the topology not in F, f(x)=0. In other words, the function f must always be zero outside F and 1 inside F, since you can chose x anywhere outside F, and yet must have f(x)=0? Or are we talking about a different continuous function f for each choice of x?

Then if f(x)=0 for all points x outside F, there seems to be a discontinuity where if you approach the edge of F from outside f=0, but if you approach the edge from inside F, then f=1. This sounds like the definition of discontinuous.

 

[dx]

We are talking about different functions for each choice of x and F.

Once you choose x and F, then we must be able to find a continuous function that separates them. This function only needs to be 0 at x, not anywhere else.

 

[WannabeNewton]

Then if f(x)=0 for all points x outside F, there seems to be a discontinuity where if you approach the edge of F from outside f=0, but if you approach the edge from inside F, then f=1. This sounds like the definition of discontinuous.

Your intuitive notions of continuity won't help here so don't use it. Even in $\mathbb{R}$ there are many textbook maps that are continuous in a counter intuitive manner. For example: http://en.wikipedia.org/wiki/Popcorn_function

Try to show this is continuous in the manner specified in the wiki article using the epsilon delta definition of continuity, it will be very instructive in allowing you to get a hold of continuity (but it isn't an easy problem so it might take some time!). In topology, continuity is not as trivial in an intuitive sense as it is for the nice functions one sees in physics.

 

[friend]

We are talking about different functions for each choice of x and F.

Once you choose x and F, then we must be able to find a continuous function that separates them. This function only needs to be 0 at x, not anywhere else.

Even if x approaches the boundary of F?

 

[dx]

Why would that be a problem? x can never be on the boundary of F since closed sets contain their boundaries, but x can be any point outside F.

 

[Bacle2]

Friend, following up upon dx's last post: have you tried to see if the result holds for F open?

Also notice, like dx said, for every choice of F and x there is a function with the given
properties; not for a given F closed and _all x_, but, given F and given a specific x not
in F . If you have a choice of F and x, you can find a function, say, f(F,x) that will have
the given properties; if you choose a different pair F', x' , you will (except maybe in very exceptional situations) a different f'(F'x').

 

[friend]

Why would that be a problem? x can never be on the boundary of F since closed sets contain their boundaries, but x can be any point outside F.

So does continuity say in other words that no matter how close you get to the boundary of F, the open nature of the entire set minus the closed set F will always allow you to find an open set even closer to the boundary that maps to an open interval even closer to 1? Though, I suppose the strict definition would require the inverse map, from the open intervals to the open sets in the topology.

 

[dx]

You can say it this way:

Since f is continuous, and since f is 1 on F, f can be made as close to 1 as you want if you evaluate it at points sufficiently close to F

In fact, in terms of neighborhoods, a map f is continuous if and only if given any point x and any neighborhood M of f(x), there is a neighborhood N of x such that f[N] is a subset of M.

 

[friend]

You can say it this way:

Since f is continuous, and since f is 1 on F, f can be made as close to 1 as you want if you evaluate it at points sufficiently close to F

In fact, in terms of neighborhoods, a map f is continuous if and only if given any point x and any neighborhood M of f(x), there is a neighborhood N of x such that f[N] is a subset of M.

Very good! Thank you. Let's move on...

A set is a closed if its complement is open, so the open sets determine what the closed sets are. And its not unique. We are not talking about some particular closed set F. For any closed set F, and any point x outside it, there must be a continuous function that separates them.

We've talked about what it means to say "any point x not in F". Now I would like to go back and understand "any closed set F". I understand that there is a whole space of points from which to arbitrarily choose any x, well, except perhaps x can not be chosen from F. But I'm not seeing how many choices we have from which to create any F.

I'm trying to avoid any appearance of uniqueness in F so that the property of complete regularity encompasses the whole topology. Otherwise, with x restricted from being chosen in F, it seems there are restricted regions for which the definition does not apply, which means the property is not applicable to the whole topology. So it seems necessary that x can be chosen from any point in the topology and we are guaranteed to be able to find an F and continuous functions f to comply with the definition. So I don't think the definition means draw a circle anywhere you like and call it F. Your comment suggests that F must be constructed from the complement of the available open sets in the topology. So it seem the topology allows more than one F so we can choose x in the previous F and still find a different F that fulfills the definition. Does this sound right?

 

[dx]

Yes, a set is closed only if its complement is an open set, i.e. a member of the topology. Given a certain closed set F, then points outside F must be separated from F by a continuous function. The points here are outside F, but they don't have to be outside other closed sets. For a different closed set F', there must be continuous functions that separate F' from points outside F', and again these points are all points outside F', whether they belong to closed sets other than F' or not.

 

[friend]

Yes, a set is closed only if its complement is an open set, i.e. a member of the topology. Given a certain closed set F, then points outside F must be separated from F by a continuous function. The points here are outside F, but they don't have to be outside other closed sets. For a different closed set F', there must be continuous functions that separate F' from points outside F', and again these points are all points outside F', whether they belong to closed sets other than F' or not.

Thank you. I'm finding it easier to understand your explanations. So is there any x anywhere in the entire topology for which we may not be able to find an F that fulfills the definition?

Does the definition also apply to subsets of the topology (subspaces?). Say we divide the topology in half. Would we be able to find x, F, and f in one half that fulfill the definition?

 

[dx]

Thank you. I'm finding it easier to understand your explanations. So is there any x anywhere in the entire topology for which we may not be able to find an F that fulfills the definition?

The definition does not say anywhere that you must be able to find an F and an x. It only says that if F is a closed set, and x is a point outside F, then there is a continuous function that separates them.

Does the definition also apply to subsets of the topology (subspaces?). Say we divide the topology in half. Would we be able to find x, F, and f in one half that fulfill the definition?

If a topological space is completely regular, then subspaces of that space are also completely regular. You could try proving that.

 

[friend]

The definition does not say anywhere that you must be able to find an F and an x. It only says that if F is a closed set, and x is a point outside F, then there is a continuous function that separates them.

If a topological space is completely regular, then subspaces of that space are also completely regular. You could try proving that.

I suspect that if a completely regular topological space can be separated into subspaces that are also completely regular, then it must have been true that in the whole topological space for any x whatsoever there must be an F and an f that fulfills the definition. For then the whole topological space could be divided into subspaces where each has a different F that allowed all x in the definition of CR in the original space.

But if you could point me to the proof of where these subspaces carry the same property, that might be worth buying one of those $100 books. Thank you.

 

[micromass]

Can we please stay on topic?

I don't like friend his behaviour either. And if it were up to me, I would have locked the thread. But dx specifically asked me to open the thread again so he could help further. I don't want this thread to turn into another shouting match.

If anybody wishes to further aid the OP, then please do so. Otherwise, I would ask to stay out of this thread.

Any off-topic posts will be deleted. Comments on this post are welcome through sending a PM or through a report.

Snarky comments will be warned/infracted. In particular, things like "Are you just guessing?" are not allowed.

 

[jbunniii]

But if you could point me to the proof of where these subspaces carry the same property, that might be worth buying one of those $100 books. Thank you.

You don't need to spend $100. Willard's General Topology covers all of this and has a retail price of US$22.95 (currently only $14.70 from Amazon). As a bonus, it contains the nice quote: "A counterexample exists showing that not every regular space is completely regular. It is formidable and we have relegated it to Exercise 18G, where most people won't be bothered by it."