What is the Significance of Completely Regular Spaces?

[WannabeNewton]

You keep thinking of this in terms of the euclidean topology on the real line. Continuity is heavily dependent upon the topology. It is not hard to come up with topologies that allow for weird continuous maps, even on the real line. How can you claim something isn't continuous without even knowing what the topology is on the set? Again, I stress that you are stuck on the euclidean topology on the real line but topology is an extremely elegant subject that generalizes continuity to a great extent.

 

[friend]

You keep thinking of this in terms of the euclidean topology on the real line. Continuity is heavily dependent upon the topology. It is not hard to come up with topologies that allow for weird continuous maps, even on the real line. How can you claim something isn't continuous without even knowing what the topology is on the set? Again, I stress that you are stuck on the euclidean topology on the real line but topology is an extremely elegant subject that generalizes continuity to a great extent.

Well I guess it's my turn... What? As I understand it, this definition concerns manifolds which are locally euclidean. So at the point on the boundary between F and not F is a point of locally euclidean geometry, right? Maybe you need to add the Hausdorff property first before if becomes a manifold. But let's assume the easiest case, how is continuity defined for f there?

 

[WannabeNewton]

The general definition of continuity for topological spaces is very simple: f:X\rightarrow Y is continuous if \forall U\subseteq Y open, f^{-1}(U) is open in X.

 

[strangerep]

PS. Next time you feel frustrated, try just simply saying, "I don't know how to make it more clear." And leave it at that. Maybe someone else will be able to explain in terms I might understand.

And maybe when someone tries to explain that you're confused about a basic essential point, perhaps you should be more receptive.

Example: in one of your earlier posts in this thread, we find the following

You are highly mistaken friend,[...] The notion of being open makes no
sense without a topology and the notion of being closed doesn't make sense
without there being a topology. Things like boundary are topological notions.
You are looking at it backwards: you cannot talk about an open set without
there being some topology not the other way around. [...]

Well, perhaps the topology was implicit in those references. I don't want to
get distracted. The question I'm concerned about at the moment is whether a
particular open set belongs to every possible topology constructed on some
background. OK , it can belong to some topology, but does it belong to every
possible topology? I would think that the fact that you can construct
different topologies from the same background means that any particular open
set does not necessarily belong to every possible topology, right?

"Distracted"??

You misunderstand the point that the term "open set" is meaningless in isolation.
Micromass, WannabeNewton, and dx have been trying to explain this to you
over and over again. But you have some kind of mental block about this (probably
because Wiki is slightly misleading on this point), hence no one can de-confuse you.

If you want to continue, please participate in this quiz:

1) What is a "topology"?

2) What is a "topological space"?

3) What is an "open set"?

4) What does "continuous" mean (in the context of general topology)?

(Answers must be given off the top of your head, not merely paraphrased from elsewhere.
"I don't know" is an acceptable answer to any or all of these.)

[BTW, @Micromass: maybe this thread should be moved into the Topology forum?]

 

[micromass]

PS. Next time you feel frustrated, try just simply saying, "I don't know how to make it more clear." And leave it at that. Maybe someone else will be able to explain in terms I might understand.

I'm not frustrated at all. My post contained genuine advice. It was not an attempt to put you down, I am merely trying to help you.
At the moment, I feel that only going through a good topology book will really help you.

 

[Bacle2]

I'm sorry you're feeling frustrated. I was doing my best to articulate the issues to try and better understand them. And I thought we were making progress. But perhaps we should move on.

I'm trying to understand the continuous nature of the map in the definition, f:X→[0,1]. It seems there is a sharp border between 0 outside and 1 inside, the edge of the closed set F. So how can it be continous?PS. Next time you feel frustrated, try just simply saying, "I don't know how to make it more clear." And leave it at that. Maybe someone else will be able to explain in terms I might understand.

I'm sorry, but I think Micromass has a point; given that many have tried unsuccessfully to help, I think is reasonable to believe your background in topology may have something to see with your trouble in understanding.

Like WBN said, there are spaces in which the topology does not allow for a visualization or representation in which the notion of sharp borders is meaningful or illustrative. Maybe the best you have is that, since F is closed , it contains its limit points so that x is not a limit point of F. In this, maybe weak sense, F and x are separated. Maybe it would help for you to consider what happens in a completely-regular space with an open set U and a point x not in U, and, in general, to consider why each part of the hypothesis is necessary in order to understand better.

 

[friend]

The general definition of continuity for topological spaces is very simple: f:X\rightarrow Y is continuous if \forall U\subseteq Y open, f^{-1}(U) is open in X.

Right, that sounds familiar. So the question would then be how is f:X→[0,1] continuous in the definition for complete regularity shown here? I'm understanding that f=0 for every x not in the closed set F, and f=1 for every y in F. I'm not seeing how one can construct open sets with just two values 0 and 1. If the set X were mapped to the continuous interval [0,1], I could see how an open set in one could be mapped to an open set in the other. But for all x in the topology, either x is in F or it is not; it seems to get mapped to either 0 or 1, and not to some continuous value in the interval. So how can a 2 valued map have open sets?

 

[friend]

I'm not frustrated at all. My post contained genuine advice. It was not an attempt to put you down, I am merely trying to help you.
At the moment, I feel that only going through a good topology book will really help you.

I appreciate your efforts. Thank you. But this whole discussion is an example of the trouble I have with topology. The presentations and books I've seen don't really show how the concepts connect to the functions one would see in physics. And I find it hard to keep it in my head because it doesn't seem relevant. However, if you know of a book that provides these connections to everyday functions and their domains and ranges, that would shed a whole new light on the subject.

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. 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.

 

[WannabeNewton]

Right, that sounds familiar. So the question would then be how is f:X→[0,1] continuous in the definition for complete regularity shown here? I'm understanding that f=0 for every x not in the closed set F, and f=1 for every y in F. I'm not seeing how one can construct open sets with just two values 0 and 1. If the set X were mapped to the continuous interval [0,1], I could see how an open set in one could be mapped to an open set in the other. But for all x in the topology, either x is in F or it is not; it seems to get mapped to either 0 or 1, and not to some continuous value in the interval. So how can a 2 valued map have open sets?

Oh boy. There are a lot of misconceptions floating around here and it isn't going to be easy to deconstruct them. I really think the best thing to do would be to properly learn topology. A forum can only do so much but I'll try to help. No one said the map had to be surjective, just that it had to be continuous and continuity does NOT mean open sets in X are taken to open sets in Y, this is an open map and you are mixing the two up. Open sets are not a property of maps so I don't know what you mean by "how can a 2 valued map have open sets?". If it is what I think it is, you are just asking how can a map take a topological space into an image containing only two values and still be continuous. Well it is trivial to come up with such maps, and even easier you can just look at constant maps which are continuous and whose image is a singleton. For example, it is very easy to show that X is a disconnected topological space if and only if there exists a non - constant continuous map f:X \rightarrow \left \{ 0,1 \right \} where the codomain of course has the discrete topology.

 

[friend]

Right, that sounds familiar. So the question would then be how is f:X→[0,1] continuous in the definition for complete regularity shown here? I'm understanding that f=0 for every x not in the closed set F, and f=1 for every y in F. I'm not seeing how one can construct open sets with just two values 0 and 1. If the set X were mapped to the continuous interval [0,1], I could see how an open set in one could be mapped to an open set in the other. But for all x in the topology, either x is in F or it is not; it seems to get mapped to either 0 or 1, and not to some continuous value in the interval. So how can a 2 valued map have open sets?

Perhaps the set in the codomain has two elements 0 and 1, a discrete set. Yet, IIRC, an open set can be constructed of discrete points, right? Then you can have open sets in [0,1] be mapped to open sets in X.

 

[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.