Exploring the Confusing Concept of Topology and Metric Space

[gotjrgkr]

Hi!

I'm a beginner for a subject "topology".
While studying it, I found a confusing concept.
It makes me crazy..
I try to explain about it to you.
For a set X, I've learned that a metric space is defined as a pair (X,d) where d is a distance function.
I've also learned that for a set X, topological space can be defined as a pair (X,T) where T indicates a topology of X.
So, I thought that a metric space is not a topological space since second coordinate of a pair (X,d) is a distance function, so that it's different from the second coordinate of the pair (X,T) where T is a topology of X.
But, here is my question, in my book, it seems that author of the book treats a metric space as a kind of topological space. For example, he defined the definition of continuity of a function from a topological space to another topological space. But, later, he uses the concept of continuity to a function from a topological space to a metric space, too.
But, since I thought that a metric space is not a topological space, it makes me confused.
So, I want to ask you if a metric space is a topological space or not.

 

[Bacle2]

Actually, every metric space (X,d) is a topological space, but not always the other way around. The basic open sets in a metric space are the open balls B(a,r)={x in X, d(x,a)<r }, meaning that s subset U of X is open if for every y in U, there is a ball B(y,r) contained in U. The simplest case, maybe of a topological space that is not metric is that of a space X with the trivial topology {X,∅}, or any non-Hausdorff topological space.

 

[gotjrgkr]

you say that every metric space is a topological space.
bUt, I'm still confused.
Let me explain it.
considering a metric space (X,d) and treating it as a topological space (X,T) where T is a topology of X, then do you mean that d = T? (that is , metric is equal to topology??)
What I know about the definitions of metric space and topological space is that they are a pair (X,d) and a pair (X,T).So, if I say a metric space is a topological space, then I think, it means that (X,d) is equal to (X,T). Then set theoritically, d = T.
But, it seems to me that this is a contradiction, isn't it??

 

[Bacle2]

No; we don't conclude d=T; that is false. A topology is a collection of open sets with certain properties, and d is just a function , so it makes no sense to say they are equal to each other. The formula d helps you determine what the open sets are. Have you read, e.g., http://en.wikipedia.org/wiki/Metric_space ?

 

[gotjrgkr]

Yeah, I've read about it...
I think I have a problem with understanding the definitions of metric space and topological space.
What I know about them is that for a set X and a metric related with the set X, the set
(X,d) is called a metric space and in the case of topological space, it is similar.
So, What I mean is that topological and metric spaces are sets (that is, a pair ( , )).
Since this is what I know about the definitions of them, I conclude the result as above.
In this definition, I can't agree with saying that a metric space is a topological space since they are different sets (as you said d is not equal to T, right?).But you told me that a metric space is a kind of topological space.
So I want to ask you what is the definition of metric or topological space that you've learned.. I want to know how you could say that a metric space is a topological space.

 

[deluks917]

Both metrics and topologies are ways of giving "structure" to a space. A metric defines a topology on a set. In this sense a metric space is a topological space. But I would not say (X,d) = (X,T) where T is the topology given by a the metric d. Two different metric spaces can have the same topology. Consider R² If |(x,y)|₁ = sqrt(x² + y²) and |(x,y)|₂ = |x| + |y| are two metrics then (R²,||₁) and (R²,||₂) are different spaces but they have the same topology.

If we only know the open sets and that they are given by a metric we cannot determine the metric. If we know the metric we know the topology. A metric gives you more information, its adds more structure to the space.

 

[gotjrgkr]

But, if metric space and topological space are defined as a pair (X,d) and (X,T), I think , it is logically obvious that (X,d) = (X,T) making a contradiction when I say the metric space is the topological space. I think I need to know about the exact definition of " structure" that you've said... I can't understand why a metric space can be treated as a topological space. I want to get an set theoritic explanation which can solve the question that I've mentioned above...

 

[Bacle2]

gotjrgkr:


A topological space (X,T) where X is a set , and T is a collection of subsets of T we call open that satisfy some specific properties. A metric space (Y,d) is a set Y , together with a function d: YxY-->ℝ⁺ union {0}. We can define a collection of subsets of Y that satisfy the properties of a topology.

 

[Bacle2]

But, if metric space and topological space are defined as a pair (X,d) and (X,T), I think , it is logically obvious that (X,d) = (X,T) making a contradiction when I say the metric space is the topological space. I think I need to know about the exact definition of " structure" that you've said... I can't understand why a metric space can be treated as a topological space. I want to get an set theoritic explanation which can solve the question that I've mentioned above...

Well, this is not a set theoretic issue, since (X,T) is not a set; it is a structure consisting of a set X and a collection T of subsets of X. And when we say (X,d) is also a topological space, we mean that we can assign to (X,d) a structure of the type (X,T). I don't see why the equality (X,T)=(X,d) would follow. At best, I can see how (X,d) is a member of the collection of all topological spaces.

 

[gotjrgkr]

Do you mean that (X,T) is just a notation to denote the "structure"??
It seems to me a little vague...
What is the exact definition of the concept "structure"??

 

[Bacle2]

Yes; note the use of the parentheses (.) and not the {.} notation, so that (X,T) denotes a pair of objects, the components of the system , and not a collection. In a collection {a,b}, the only thing that we consider is membership; in a structure, we consider the components and the rules guiding their interaction.


More generally, a structure, I guess, would be/is equivalent to a system; a collection of objects that interact in well-defined or clearly-defined ways, e.g., a country could be described ( in an oversimplified way) as a structure consisting of a collectionX of people, and a system of government/laws , a culture ,etc.. A car is a structure consisting of a physical object, an underlying electrical, mechanical system, together with the rules that guide/describe the interaction between the components of the structure. Of course you can emphasize or deemphasize certain components if you choose or need to. So, in our case (X,T) is a structure, where the components are X and T. The rules describing their interaction is/are, basically, the concept of topology. Similarly, (X,d) is a structure consisting of the set X ofobjects and d is a function satisfying specific properties, that assigns to a pair of objects a nonnegative real number. The rules guiding/describing their interaction is the area of metric spaces.

 

[gotjrgkr]

Yes; note the use of the parentheses (.) and not the {.} notation, so that (X,T) denotes a pair of objects, the components of the system , and not a collection. In a collection {a,b}, the only thing that we consider is membership; in a structure, we consider the components and the rules guiding their interaction. More generally, a structure, I guess, would be/is equivalent to a system; a collection of objects that interact in well-defined or clearly-defined ways, e.g., a country could be described ( in an oversimplified way) as a structure consisting of a collectionX of people, and a system of government/laws , a culture ,etc.. A car is a structure consisting of a physical object, an underlying electrical, mechanical system, together with the rules that guide/describe the interaction between the components of the structure. Of course you can emphasize or deemphasize certain components if you choose or need to. So, in our case (X,T) is a structure, where the components are X and T. The rules describing their interaction is/are, basically, the concept of topology. Similarly, (X,d) is a structure consisting of the set X ofobjects and d is a function satisfying specific properties, that assigns to a pair of objects a nonnegative real number. The rules guiding/describing their interaction is the area of metric spaces.

um... I see...
It is still a little vague for me. I, however, got it intuitively..
I've got an another question..
I've learned that a concept of continuity for a function from a topological space into another topological space is defined.
More specifically (as you know), let (X,T), (Y,U) be two topological spaces.
then for a function f mapping of X into Y, f is continuous $\Leftrightarrow$
for each open subset V of Y, the set f$^{-1}$(V) is an open subset of X.

If I admit that a metric space (X,d) is a topological space, then can I also define a notion of continuity for a function from a topological space (Z,L) into a metric space(X,d) by using above definition??
Then what is the definition??

 

[Fredrik]

For a set X, I've learned that a metric space is defined as a pair (X,d) where d is a distance function.
I've also learned that for a set X, topological space can be defined as a pair (X,T) where T indicates a topology of X.
So, I thought that a metric space is not a topological space since second coordinate of a pair (X,d) is a distance function, so that it's different from the second coordinate of the pair (X,T) where T is a topology of X.
But, here is my question, in my book, it seems that author of the book treats a metric space as a kind of topological space. For example, he defined the definition of continuity of a function from a topological space to another topological space. But, later, he uses the concept of continuity to a function from a topological space to a metric space, too.
But, since I thought that a metric space is not a topological space, it makes me confused.
So, I want to ask you if a metric space is a topological space or not.

Strictly speaking, it's not. But you probably know that when (X,T) is a topological space, it's standard to call X a topological space. This is a bit sloppy, but it doesn't confuse anyone who understands the definitions. Similarly, when (X,d) is a metric space, it's standard to call X a metric space. So if you have defined both a topology and a metric on the same set X, this standard abuse of terminology allows you to say that X is both a topological space and a metric space.

Now let's see what we can say without abusing the terminology. Suppose that (X,d) is a metric space. For each x in X, and each r>0, define the open ball of radius r around x as the set $B(x,r)=\{y\in X|d(x,y)<r\}$. Define $\tau_d$ as the set of all subsets of X that are equal to a union of open balls. Now $\tau_d$ is a topology on X, called the metric topology associated with the metric d, and $(X,\tau_d)$ is a topological space.

So (X,d) isn't a topological space, but there's always a topological space with the same underlying set X, and a topology $\tau_d$ determined by the metric d. This is what mathematicians mean when they say that a metric space "is" a topological space.

What is the exact definition of the concept "structure"??

A definition that's general enough to cover groups, vector spaces, topological spaces, metric spaces, smooth manifolds etc. would be very long and awkward, and not of much use to you here. So I suggest that you don't worry about that and just try to understand roughly what term refers to. Loosely speaking, it's "a set and then some". In the case of a metric space, it's the pair (X,d). In the case of a vector space, it's the triple (X,A,S) where A is the addition operation and S is the scalar multiplication operation. Typically, a structure is an n-tuple where the first element is a set called the underlying set (or the domain or the universe) of the structure, and the other elements are relations (like <) or operations (like addition or multiplication) on X.

let (X,T), (Y,U) be two topological spaces.
then for a function f mapping of X into Y, f is continuous $\Leftrightarrow$
for each open subset V of Y, the set f$^{-1}$(V) is an open subset of X.

If I admit that a metric space (X,d) is a topological space, then can I also define a notion of continuity for a function from a topological space (Z,L) into a metric space(X,d) by using above definition??
Then what is the definition??

You would apply that definition to the topological space $(X,\tau_d)$, where $\tau_d$ is the metric topology associated with d.

 

[gotjrgkr]

I see... I also want to believe that it is true.
I've been actually studying a book "Topology" written by author munkres..
In his book, if I follow your opinion, there's a part which I can't interpret.
He defined a subspace only for a topological space. It means that he doesn't define a concept of a subspace for a metric space if I agree that strickly speaking, a metirc space is not a topological space. When I see a statement " Subspaces of metric spaces behave the way one would wish them to" in p.129 of his book with your opinion, then it's confusing for me to interpret it since a metric space is different from a topological space and a concept of a subspace is defined only for a topological space. That is, if I try to interpret the meaning "subspace of a metric space", I think that I should treat the metric space as a topological space...
How do you think about this point that I've mentioned above?

Ah, and in addition to the above question, what textbook do you use for studying topology??
Could you recommend a good one for me if you know??

 

[Fredrik]

I see... I also want to believe that it is true.
I've been actually studying a book "Topology" written by author munkres..
In his book, if I follow your opinion, there's a part which I can't interpret.
He defined a subspace only for a topological space. It means that he doesn't define a concept of a subspace for a metric space if I agree that strickly speaking, a metirc space is not a topological space. When I see a statement " Subspaces of metric spaces behave the way one would wish them to" in p.129 of his book with your opinion, then it's confusing for me to interpret it since a metric space is different from a topological space and a concept of a subspace is defined only for a topological space. That is, if I try to interpret the meaning "subspace of a metric space", I think that I should treat the metric space as a topological space...
How do you think about this point that I've mentioned above?

Ah, and in addition to the above question, what textbook do you use for studying topology??
Could you recommend a good one for me if you know??

Suppose that (X,d) and (Y,d') are metric spaces. (Y,d') is said to be a subspace of (X,d) if Y is a subset of X, and d' is the restriction of d to Y×Y. This is consistent with the definition of a subspace of a topological space in several different ways, for example the metric topology on Y associated with d' coincides with the topology on Y inherited from $(X,\tau_d)$ where $\tau_d$ is the metric topology associated with d.

I learned pieces of topology from several different sources, so I don't have a specific book to recommend. I suggest that you search the science book forum for threads about it.

 

[gotjrgkr]

So, since a metric space is basically different from a topological space(I mean that a metric space is not a special kind of a topological space), a topological space is not a expanded notion of a metric space?? Do I have to deal with a topological space separately from the notion of a metric space?
I can't understand why some books say topological space is related with a metric space in that it is more expanded notion since it also deals with a continuity for a function over spaces in which any distance function doens't exist...

 

[Bacle2]

A metric space is precisely a particular type of topological space, so that every metric space is a topological space, but not the other way around. In a metric space, the topology is generated, in a sense, by the distance function. I would suggest you may be being too intense about getting an in-depth understanding at this early point; don't get me wrong, I do endorse the idea of aiming for in-depth understanding; only you may have to do more exercises early on, think a little less, and the in-depth understanding will likely come in time.

 

[Fredrik]

So, since a metric space is basically different from a topological space(I mean that a metric space is not a special kind of a topological space), a topological space is not a expanded notion of a metric space??

Since every metric space defines a topological space in a standard way, I think it's very natural to think of metric spaces as a special kind of topological space (a "metrizable" topological space), even though you can't put an equality sign between (X,τ) and (X,d).

You need to understand that everyone abuses the terminology somewhat, even in theorems and definitions. For example consider this:

Suppose that $(X,\tau_X)$ and $(Y,\tau_Y)$ are topological spaces. A function $f:X\to Y$ is said to be continuous if $f^{-1}(E)$ is open for every open $E\subset X$.​

This is a typical definition that you might find in a topology book, but if you don't realize that it assumes that you understand that it's abusing the terminology, it will (or should) look like nonsense. This would be the "non-abusive" way to state it:

Suppose that $(X,\tau_X)$ and $(Y,\tau_Y)$ are topological spaces. A function $f:X\to Y$ is said to be continuous with respect to the pair of topologies $(\tau_X,\tau_Y)$ if $f^{-1}(E)$ is open with respect to $\tau_X$ for every $E\subset X$ that's open with respect to $\tau_Y$.​

Theorems and definitions get pretty long and awkward if we specify the relevant topologies and metrics in every statement, so no one does. I don't think there's a topology book that doesn't abuse the terminology to some degree.

Do I have to deal with a topological space separately from the notion of a metric space?

Yes, because some topological spaces aren't metrizable. (But a vast majority of the interesting ones are). Theorems that hold for metric spaces automatically hold for metrizable topological spaces, but some of them don't hold for non-metrizable topological spaces.

I can't understand why some books say topological space is related with a metric space in that it is more expanded notion since it also deals with a continuity for a function over spaces in which any distance function doens't exist...

They do because all metric spaces "are" topological spaces in the sense explained in my previous post. Also, all definitions in the context of topological spaces, of terms that already had a definition in the context of metric spaces, are stated so that the new definition agrees with the old in the context of metrizable topological spaces. "Continuous" is a good example of that.

 

[micromass]

Saying that a metric space is a topological space is false. It is better to say that a metric space uniquely determines a topological space. So with every metric space, I can associate one and only one topological space.
Many authors however, do not distinguish between a metric and a topological space. They will say that a metric space IS a topological space. Whenever you read that, you must know that this is not true, and that they actually mean that a metric space determines a topological space!

 

[Bacle2]

I don't think it is false to say a metric space is a topological space; it may be loose talk, but it depends on the level and type of structure you are considering. I think from this conversation/thread, it is clear that it means that a metric space can be given the structure of a topological space that somehow agrees with the distance function. If you drew a Venn diagram, with a bubble containing all the topological spaces, the collection of metric spaces would be contained in the bubble of topological spaces.

 

[micromass]

I don't think it is false to say a metric space is a topological space; it may be loose talk, but it depends on the level and type of structure you are considering. I think from this conversation/thread, it is clear that it means that a metric space can be given the structure of a topological space that somehow agrees with the distance function. If you drew a Venn diagram, with a bubble containing all the topological spaces, the collection of metric spaces would be contained in the bubble of topological spaces.

Well, it is confusing the OP. So I just want to make clear that when working strictly logical, that a metric space is not a topological space. Saying that a metric space is a topological space is an abuse of language (an abuse that everybody does though).

 

[gotjrgkr]

In my book, there are two definitions(Every theroem and definition which follow are referred to a book "topology" by author munkres);
Def.1
: A collection A of subsets of a space X is said to cover X, or to be a covering of X, if the union of the elements of A is equal to X. It is called an open covering of X if its elements are open subsets of X.

Def.2
: A space X is said to be compact if every open covering A of X contains a finite subcollection that also covers X.

So, compactness is defined for a (topological) space.

In this book, there's a theorem like this;

Th.27.6(Uniform continnuity theorem).
Let f : X$\rightarrow$Y be a continuous map of the compact metric space (X,d$_{X}$) to the metric space (Y,d$_{Y}$). Then f is uniformly continuous.

From above Theorem, there are two problems for me. First of all, the continuity is defined only for a function over two topological spaces, so that if I follow an opinion which says a metric space is logically different from a topological space as micromass said, I can't get an idea why the "metric space" is in the above theorem. I think, those positions are not suitable for the metric spaces.
Second, if a metric space is basically different from a topological space, then what does the "compact metric space" mean?? In the same aspect from the above fist problem, I think the word "metric space" is not suitable for the position written.

If I follow (micromass or frederik)'s opinion, then how can you explain about my problems??

 

[micromass]

In my book, there are two definitions(Every theroem and definition which follow are referred to a book "topology" by author munkres);
Def.1
: A collection A of subsets of a space X is said to cover X, or to be a covering of X, if the union of the elements of A is equal to X. It is called an open covering of X if its elements are open subsets of X.

Def.2
: A space X is said to be compact if every open covering A of X contains a finite subcollection that also covers X.

So, compactness is defined for a (topological) space.

In this book, there's a theorem like this;

Th.27.6(Uniform continnuity theorem).
Let f : X$\rightarrow$Y be a continuous map of the compact metric space (X,d$_{X}$) to the metric space (Y,d$_{Y}$). Then f is uniformly continuous.

From above Theorem, there are two problems for me. First of all, the continuity is defined only for a function over two topological spaces, so that if I follow an opinion which says a metric space is logically different from a topological space as micromass said, I can't get an idea why the "metric space" is in the above theorem. I think, those positions are not suitable for the metric spaces.
Second, if a metric space is basically different from a topological space, then what does the "compact metric space" mean?? In the same aspect from the above fist problem, I think the word "metric space" is not suitable for the position written.

If I follow (micromass or frederik)'s opinion, then how can you explain about my problems??

A function $f:X\rightarrow Y$ between metric spaces induces a function between topological spaces. We call a function between metric spaces continuous if the induced function between topological spaces is continuous.

A metric space is compact if the induced topological space is compact.

You will have to make all these translations for yourself now. The author will always assume now that a metric space IS a topological space. This is an abuse of notation, but everybody does it.

 

[gotjrgkr]

A function $f:X\rightarrow Y$ between metric spaces induces a function between topological spaces. We call a function between metric spaces continuous if the induced function between topological spaces is continuous.

A metric space is compact if the induced topological space is compact.

You will have to make all these translations for yourself now. The author will always assume now that a metric space IS a topological space. This is an abuse of notation, but everybody does it.

Ah...!
Ok...
Then, how about this function f: X$\rightarrow$Y where X is a topological space and Y is a metric space?
Does it then mean that f is continuous if a function f relative with the topological space X and a topological space indeced by the metric space Y is continuous??
connectedness also can be defined for a metric space??
Is it right for me to say that a metric space is connected if the topological space induced by the metric space is connected??
How can you explain to me about a subspace of a metric space??
In p. 129 of munkres's book, there'is a following statement;
Subspaces of metric spaces behave the way one would wish them to.
Since I've learned the definition of a subspace of a topological space only, I was confused to see the meaning of a subspace of a metric space. As you said, a metric space is differrent from a topological space, right??
I know that replying all of my questions above would be very laborious work for you, but could you explain it for me please,?

 

[Fredrik]

Then, how about this function f: X$\rightarrow$Y where X is a topological space and Y is a metric space?
...
How can you explain to me about a subspace of a metric space??

I answered these two in posts #13 and #15 respectively. f is continuous with respect to the topology on X and the metric on Y, if $f^{-1}(E)$ is open with respect to the topology on X whenever E is open with respect to the metric on Y. A subspace of a metric space can be obtained by restricting the domain of the metric to Y×Y, where Y is a subset.

connectedness also can be defined for a metric space??
Is it right for me to say that a metric space is connected if the topological space induced by the metric space is connected??

Yes.

 

[gotjrgkr]

Thanks for all of you!
It really helps me a lot!
Thanks!