Generalizations (from metric to topological spaces)

[Fredrik]

This is kind of a weird question. I like to think about how I would explain things to other people, and I realized that I don't know a great way to explain in general how terms defined in the context of metric spaces are generalized to the context of topological spaces. It's not at all difficult to generalize a term if you already know how to state the original definition in terms of open sets. For example, this is one way to define the term "closed" in the context of metric spaces:

Suppose that $(X,d)$ is a metric space. A set $E\subset X$ is said to be closed if $E^c$ is open.​

All we have to do to generalize the term to the context of topological spaces is to change the first sentence, and leave everything else word-for-word the same:

Suppose that $(X,\tau)$ is a topological space. A set $E\subset X$ is said to be closed if $E^c$ is open.​

The only problem with this approach is that it might be possible to state the original definition in terms of open sets in more than one way. "Closed" can be equivalently defined like this:

Suppose that $(X,d)$ is a metric space. A set $E\subset X$ is said to be closed if the limit of every convergent sequence in E is in E.​

(This can be thought of as a definition in terms of open sets, since we can define limits of sequences in terms of open sets). If we apply the generalization procedure discussed above to this version of the definition, what we end up with is something that certainly deserves to be called a generalization of the term "closed". But it wouldn't be equivalent with the first generalization we found. The first mathematicians who figured this out had to choose between at least three options: 1) use the first one, 2) use the second one, 3) don't generalize the term at all; instead assign different terms to the concepts associated with the two possible generalizations, for example "topologically closed" and "sequentially closed".

The consensus today is to use option 1, and to use the term "sequentially closed" for the concept associated with the second generalization. This brings me to what I want to ask. I understand why option 1 is the preferred generalization of "closed". However, it seems to me that there are always two generalizations, one that involves limits of sequences and one that doesn't, and it seems to me that we always end up choosing the one that doesn't involve sequences. So I'm wondering if there's a way to argue for this in general. Is there a way to know that the generalization that doesn't involve sequences will always be the more useful one, or did the mathematicians who chose the generalizations have to consider each term separately? Is there an example of a term for which the generalization that involves sequences has become the standard?

 

[micromass]

Hmm, good question. My guess is that the early topologists really had the choice between two definition and they had to check some properties before they encountered the right one.

It is interesting to note that the first approach to an axiomatization of topology happened by defining limit points axiomatically. This was done by Riesz in a paper of 1909 that I'm now going to search and read. The more common open set definitions are due to Hausdorff.

My guess is that Riesz first tried to set axioms for sequences, but I think that he soon found out that this approach didn't work. So in a way, he was forced to consider limit points of sets. This would give him the notion of topological space that generalizes easiest.

 

[Bacle]

This is kind of a weird question. I like to think about how I would explain things to other people, and I realized that I don't know a great way to explain in general how terms defined in the context of metric spaces are generalized to the context of topological spaces. It's not at all difficult to generalize a term if you already know how to state the original definition in terms of open sets. For example, this is one way to define the term "closed" in the context of metric spaces:

Suppose that $(X,d)$ is a metric space. A set $E\subset X$ is said to be closed if $E^c$ is open.​

All we have to do to generalize the term to the context of topological spaces is to change the first sentence, and leave everything else word-for-word the same:

Suppose that $(X,\tau)$ is a topological space. A set $E\subset X$ is said to be closed if $E^c$ is open.​

The only problem with this approach is that it might be possible to state the original definition in terms of open sets in more than one way. "Closed" can be equivalently defined like this:

Suppose that $(X,d)$ is a metric space. A set $E\subset X$ is said to be closed if the limit of every convergent sequence in E is in E.​

(This can be thought of as a definition in terms of open sets, since we can define limits of sequences in terms of open sets). If we apply the generalization procedure discussed above to this version of the definition, what we end up with is something that certainly deserves to be called a generalization of the term "closed". But it wouldn't be equivalent with the first generalization we found. The first mathematicians who figured this out had to choose between at least three options: 1) use the first one, 2) use the second one, 3) don't generalize the term at all; instead assign different terms to the concepts associated with the two possible generalizations, for example "topologically closed" and "sequentially closed".

The consensus today is to use option 1, and to use the term "sequentially closed" for the concept associated with the second generalization. This brings me to what I want to ask. I understand why option 1 is the preferred generalization of "closed". However, it seems to me that there are always two generalizations, one that involves limits of sequences and one that doesn't, and it seems to me that we always end up choosing the one that doesn't involve sequences. So I'm wondering if there's a way to argue for this in general. Is there a way to know that the generalization that doesn't involve sequences will always be the more useful one, or did the mathematicians who chose the generalizations have to consider each term separately? Is there an example of a term for which the generalization that involves sequences has become the standard?

Doesn't a closed set generalize as a set that contains all its limit points (tho, as you said,
limit points may not always be determined by limits of sequences)?

Still, you may want to look at sequential spaces: http://en.wikipedia.org/wiki/Sequential_space

I don't know much about them. Still, it is interesting too, and I think it relates to your

post --sorry otherwise -- how one decides what notion of convergence to use in

spaces that don't have a "natural" notion, like, e.g., that of the space of distributions,

i.e., the dual space of bump functions (C^oo -functions :R^n-->R with compact support ),

where one uses sequences to determine continuity (and this is decided before assigning

a topology to the space of distributions). Anyway, I hope that is not too much ranting.

 

[Fredrik]

Thanks guys. I have to get some sleep now. I'll write a better reply tomorrow.

 

[lavinia]

While the idea of a topological space is more general than the idea of a metric space, it seems that once one realizes that the properties of continuous functions on a metric space are completely determined by the way they handle open sets, one naturally arrives at the the idea of a topology and no longer has to think in terms of Cauchy sequences. To me this is a profound insight for metric spaces. For instance in learning about holomorphic functions one cares about their mapping properties, not how they handle Cauchy sequences.After this insight on metric spaces one may ask whether this idea is useful for other spaces as well.

 

[Fredrik]

While the idea of a topological space is more general than the idea of a metric space, it seems that once one realizes that the properties of continuous functions on a metric space are completely determined by the way they handle open sets, one naturally arrives at the the idea of a topology and no longer has to think in terms of Cauchy sequences.

Cauchy sequences? They don't seem to be particularly relevant to what I'm talking about. I'm talking about how most terms in the context of metric spaces have definitions in terms of limits of sequences, and how this often gives us two non-equivalent ways to generalize the term we're concerned about.

After this insight on metric spaces one may ask whether this idea is useful for other spaces as well.

I've been thinking something similar to this, but I've been using the concept of limits of sequences as the starting point. The usual definition of a limit of a sequence in the context of metric spaces is equivalent to "x is said to be a limit of a sequence if every open set that contains x contains all but a finite number of terms". This means that if we know which sets to call "open", we can determine if a given sequence is convergent, and find its limit if it has one, without even knowing the metric. This observation is what I've been thinking of as the primary motivation for topological spaces. (Of course, it doesn't fully explain why the definition of "topological space" includes those axioms and no others, so maybe I can do better).

Doesn't a closed set generalize as a set that contains all its limit points (tho, as you said,
limit points may not always be determined by limits of sequences)?

This is a good point. This gives us a third way to generalize the term "closed". It turns out to be equivalent to the first generalization I suggested, but that's not obvious until we've proved it.

Another reason I'm glad you mentioned this is that I think this definition of closed set may be a nice way to introduce nets. In the context of metric spaces, it's really easy to see that this definition of "closed" implies that if E is closed and x is in E, there's a sequence in E that converges to E. (Define the sequence by choosing one term in B(x,1/n) for each n). In the context of topological spaces, we can do essentially the same thing. We just have to consider a countable subset of the set of open neighborhoods of x, and choose one term in each open neighborhood of x that belongs to that subset. But we can also choose to define something new (to be called a "net") by choosing a "term" in every open neighborhood of x, and then explain how this generalizes the concept of a sequence, how there's a natural way to define convergence, etc.

Still, you may want to look at sequential spaces: http://en.wikipedia.org/wiki/Sequential_space

That might be useful. I will think about it.

Still, it is interesting too, and I think it relates to your

post --sorry otherwise -- how one decides what notion of convergence to use in

spaces that don't have a "natural" notion, like, e.g., that of the space of distributions,

i.e., the dual space of bump functions (C^oo -functions :R^n-->R with compact support ),

where one uses sequences to determine continuity (and this is decided before assigning

a topology to the space of distributions).

I brought this up in this thread. (I'm guessing that you got it from the Wikipedia article that I linked to in post #3). Micromass said (in post #4) that this looks wrong to him.

My guess is that the early topologists really had the choice between two definition and they had to check some properties before they encountered the right one.

Yes, the more I think about it, the more it seems that way to me. A google search I did yesterday (I don't remember what I was searching for) took me to the "historical notes" part of Willard. There are some interesting comments about the generalization of "compact" in the notes on section 17, starting near the bottom of page 303. It looks like they just compared the possible generalizations of that particular term and found that one of them had a nice property that the other one didn't.

 

[Bacle]

Yes, sorry, Fredrik, this is the source
I really wanted to quote:

http://planetmath.org/encyclopedia/TopologyViaConvergingNets.html