Question on Topological Spaces

**lethe** · Feb27-04, 01:40 PM

Originally posted by scott
The way I understand it, a topological space T, is defined as a subset of a set X

T is a subset of the power set of X, not of X itself.

**scott** · Feb27-04, 02:22 PM

I mistyped above, I meant to say that the set X is the topological space. I don't think I'm that far off, I realize now that X is the topological space which has the set T, a subset of the power set of X, as a topology on it.

Forgive my non-rigorous, non-formal language, but I picture a subset T of a set X as "residing within" X. Maybe that's not the right description, but I often see terminology such as "for T IN X" which I would understand as, contained within, and element of, or residing within. Those phrases are all equivalent to me. Is that wrong?

**matt grime** · Feb27-04, 02:46 PM

I think all those statements are ok, but you are using T for two different things. T as a topology - a collection of subsets of X, and T as an element of the topology.

It is ok, in a non-rigorous sense, to say that an element of the toplogy resides in the space, meaning it is a subset of it, but it is wrong to say that a topology resides in the set X in this sense, I think that just comes down to a reasonable sense of semantics.

Just differentiate between the topology T on a set X, and an element of the topology. In your terms, each element of T resides in X, but T does not reside in X. More precisely every element of T is a subset of X, but the collection T is not a subset of X.

**matt grime** · Feb27-04, 02:51 PM

Originally posted by scott
in Euclidean space with a traditional x,y Cartesian graph, you have sets of points which are defined by a distance function between them, and this distance function is a Metric on the space, i.e., the set of points defined by the Metric resides within the Euclidean space, sort of the way the T subsets satisfying the properties are a topology on a topological space X.

No, the points are not defined by a distance function at all. Euclidean 2-space exists without a metric.

You can use the distance function to define the open sets in a topology on R^2, which agrees with the product topology on R^2 - that's an interesting elementary problem for you to prove.

**NateTG** · Feb27-04, 04:45 PM

Scott:

Let's say I have some set

, and it's got the power set

.

Now, let's sayt that $LaTeX Code: \\tau \\subset P$ . Then the elements $LaTeX Code: i \\in \\tau$ are subsets of

.

We say that $LaTeX Code: \\tau$ is a topology on

if:
1. $LaTeX Code: \\tau$ contains

.
2. $LaTeX Code: \\tau$ contains the empty set.
3. $LaTeX Code: \\tau$ is closed over arbitrary unions.
4. $LaTeX Code: \\tau$ is closed over finite intersections.

Here are some examples of topolgies that might be helpfull:

For any

, $LaTeX Code: \\tau=\\{X,\\{\\}\\}$ is a topology on X, and $LaTeX Code: \\tau=2^X$ is a topology on

. Neither of these is usualy particularly useful or interesting. You might want to check that these are topologies.

For $LaTeX Code: X=\\{0,1,2,3,4\\}$ , $LaTeX Code: \\tau=\\{ \\{\\},\\{1\\},\\{0,1,2,3,4\\}\\}$ is a topology on

.

On the reals, there are many different potential topologies. Other posters have already mentioned the Zariski topology where for any $LaTeX Code: t \\in \\tau$ ,

is the empty set, or $LaTeX Code: \\Re - t$ is finite.

There's the usual topology on the reals $LaTeX Code: \\tau$ is the set of all sets

with the property that $LaTeX Code: \\forall x \\in t \\exists \\epsilon > 0 s.t. (x-e,x+e) \\subset t$ .

The lower limit topolgy replaces that requirement with:
$LaTeX Code: \\forall x \\in t \\exists \\epsilon > 0 s.t. [x,x+e) \\subset t$ .

It's easy to see that other requirements also work:
$LaTeX Code: \\forall t, x \\in t \\rightarrow (x+1) \\in t$
or
$LaTeX Code: \\forall t, x \\in t \\rightarrow x^2 \\in t$
are also toplogies on $LaTeX Code: \\Re$

The sets $LaTeX Code: t \\in \\tau$ are referred to as open sets. For example, in the lower limit topology,

is an open set, but it is not an open set in the usual topolgy on the reals.

Now, a topological space

is a set

and a topolgy on

, $LaTeX Code: \\tau$ .

Because "in" is usually used to refer to set inclusion in mathematics i.e. $LaTeX Code: \\in$ , it's a bad idea to use it for anything else. You can say that $LaTeX Code: \\tau$ is a topology on

. Since most sets have more than one topology, it's also poor form to say that $LaTeX Code: \\tau$ is the topology on

.

**scott** · Mar2-04, 12:15 PM

Ok thanks for the replies guys. One thing I've been struggling with a little is the idea of moving from the specific to the general in going from definitions of Euclidiean to Metric, to Topological spaces. I think I may be on the right track. Here's a specific example to illustrate going from the least general idea to the most general idea, tell me if I'm off here:

Given two sets, X and Y, the Cartesian product of the two sets forms an n-tuple of ordered pairs (x,y) for all x in X and all y in Y. The collection of all possible ordered pairs (x,y) forms the basis of a space. This collection of all possible ordered pairs of (x,y) forms a set, we'll call

. If the space is Euclidean then

is the set of real numbers together with a distance function containing certain properties (in this case the Pythagoean theorem).

To further generalize the idea of the space we can talk about a Metric space. Using the above example, the metric space is again the set

, and has the same properties as the Euclidean space, except that

isn't limited to the set of real numbers, and the distance function doesn't necessarily have to be the Pythagorean theorum.

To generalize even further, we can move to the idea of a topological space. Using the above example, the topological space is again the set

, except this time we aren't concerned with a distance function. Rather, we look only at the collection of sets and subsets containing certain properties that can be constructed on

. A collection of subsets T, containing certain properties, which can be constructed on

is said to be a topology on

, and

is a topological space.

How's that? Not real formal or complete, but I'm just trying to convey an understanding of the concepts involved in the relationships of these spaces to eachother.

**matt grime** · Mar2-04, 01:34 PM

in order hopefully

(x,y) is a 2-tuple. the collection of all ordered pairs is XxY, why invent a new terminology that doesn't even mention Y? basis has a different meaning than this, perhaps you want base, but that isn't necessarily a good word either.

what do you mean, Euclidean space? R, R^2, etc are all euclidean (with the Euclidean metric - there are other metrics on them)

I don't understand why the idea of caretesian product arose. when i said a topological space was a pair (X,T) i didn't mean a Cartesian product as you've got it there.

**scott** · Mar2-04, 01:50 PM

(x,y) is a 2-tuple. the collection of all ordered pairs is XxY, why invent a new terminology that doesn't even mention Y? basis has a different meaning than this, perhaps you want base, but that isn't necessarily a good word either.

I'll double-check my use of those terms again.

I meant R^2 and was using that as an example with which to demonstrate the relationship between different types of spaces all using that example.

I understand that the Pair you referred to (X,T) is not the ordered pair in my example. Again, I was using the ordered pairs in the example of R^2 space as a set which forms a space. That set would be the X in your pair (X,T). In looking at the space as a topological space, we are only concerned with a collection of particular subsets T, and the properties thereof, as opposed to defining any distance formula as we would do if we are looking at the space as a metric space.

Again, I was not trying to formally define anything, just comparing and contrasting the different concepts of spaces using a single example.

**NateTG** · Mar2-04, 03:19 PM

Originally posted by scott

Given two sets, X and Y, the Cartesian product of the two sets forms an n-tuple of ordered pairs (x,y) for all x in X and all y in Y.

The cartesian product is not an n-tuple. The cartesian product is a set of ordered pairs.

The collection of all possible ordered pairs (x,y) forms the basis of a space.

"Basis" is a topological term with a specific meaning, and I doubt you mean to use it in that way.

This collection of all possible ordered pairs of (x,y) forms a set, we'll call .

Why confuse everyone (probably including yourself) when you can use $LaTeX Code: X \\times Y$ instead? Everyone who recognizes that

and

are sets will recognize that as their cartesian product.

If the space is Euclidean then is the set of real numbers together with a distance function containing certain properties (in this case the Pythagoean theorem).

Euclidean spaces are of the form $LaTeX Code: \\Re^n$ . The distance function is not a necessary aspect of a euclidean space. Also,

is not known to be the set of real numbers ( $LaTeX Code: \\Re^2$ is euclidean but not equal to $LaTeX Code: \\Re$ ), and it's better to say a "distance function" or "function $LaTeX Code: d:X_x \\times X_x \\rightarrow \\Re$ having certain properties" than using "containing". Since I expect that you mean the normal requirements for distance functions, 'certain properties' should be removed, and filed with the department of redundancy department. It also implies that

's additional properties will be described later.

To further generalize the idea of the space we can talk about a Metric space.

It's not possible to generalize an single instantiation, either you're generalizing the notion of space, or you're talking about "the space" which you've (in theory) previously described or defined. Not to mention that you haven't really discussed a notion of space, or generalized it, so 'further' is inappropriate.

Using the above example, the metric space is again the set , and has the same properties as the Euclidean space, except that isn't limited to the set of real numbers, and the distance function doesn't necessarily have to be the Pythagorean theorum.

This paragraph is essentially gibberish.

I recommend you forget about metric spaces until you get the hang of basic point set topology. Munkres (not sure about the spelling) is an excellent introductory text.

You don't need to get into metric spaces, or cartesian products in order to demonstrate your understanding of topology.

Here's excercises for you:

List all of the topolgies on the set $LaTeX Code: X=\\{0,1\\}$ . (A topolgy is a complete list of open sets.)

Show that there are at least 45 different topolgies on the set:
$LaTeX Code: X=\\{0,1,2\\}$

For $LaTeX Code: X=\\{0,1,2\\}$ give an example of a subset

of

that contains the

and the empty set but is not a topology on

.

**scott** · Mar2-04, 06:31 PM

The Cartesian product is not an n-tuple. The cartesian product is a set of ordered pairs.

My bad. I'll re-read the definitions.

"Basis" is a topological term with a specific meaning, and I doubt you mean to use it in that way.

ditto

Why confuse everyone (probably including yourself) when you can use instead? Everyone who recognizes that and are sets will recognize that as their cartesian product.

Just about every place I've seen defines the set of ordered pairs that's way, that's why.

here's a snippet from a mulivariable calculus book:

Suppose A and B are sets. The
Cartesian product AxB of these sets is the collection of all ordered pairs (a,b) such that a is in A and b is in B.

I can find many other examples of the Cartesian product defined in just this way. I was merely expressing it the way I've seen it.

As for the rest, I keep seeing all different kinds of spaces described and defined in various sources. A lot of the terms keep popping up over and over. Why do we need all these different kinds of spaces? Why not just say "space"? There must be some difference or distinction between the different kinds of spaces, and there must be some reason for distinguishing them.

It helps me in attempting to understand what these different kinds of spaces are and how do distinguish between them by comparing and contrasting them, especially because I've read, for example, that a Euclidean space is a metric space and a topological space. So I tried to find an example that was all three and distinguish between the different concepts of space.

As I've said repeatedly, I'm not attempting a formal definition or complete treatment of these topics, only an exercise to help me understand the distinction between them.

Also, I don't appreciate my comments being called "gibberish". This strikes me as pedantic and snobbish, and not in any way instructive or representative of a genuine effort to help me understand.

**matt grime** · Mar2-04, 06:50 PM

the product notation thing was 'cos you labelled the product of X and Y X_x, which isn't standard - where's the Y gone?

**NateTG** · Mar2-04, 07:05 PM

Regarding Cartesian products:

The elements of a cartesian product are n-tuples, but the product itself is a set.

Regarding 'euclidean' spaces, metric spaces:

Euclidean space is $LaTeX Code: \\Re^n$ , for some n. The ordered n-tupes in $LaTeX Code: \\Re^n$ can also be regarded as vectors.

If your course is in point set topology, then you should probably not be dealing with metric spaces quite yet. The euclidean metric on $LaTeX Code: \\Re^n$ is $LaTeX Code: d(\\vec{x},\\vec{y})=|(\\vec{x}-\\vec{y})|$ where the vertical bars represent the vector norm.

My appologies regarding the gibberish comment.

**NateTG** · Mar2-04, 07:45 PM

Metric spaces are a type of topology, where all open sets

have the property that $LaTeX Code: \\forall x \\in O \\exists \\delta s.t. d(x,y) < \\delta \\rightarrow y \\in 0$

There are other ways to describe metric spaces, but the only one that I am familiar with involves the notion of basis.

**scott** · Mar11-04, 12:19 PM

I'm not actually in a course. Just a soul who's insatiably curious about the universe and how it works. My intermediate goal is to understand the mathematics of general relativity. The most formal instruction I've had in math would be up through introductory calculus. I've gone back and reviewed basic algebra, trig, and calculus and now I'm going through multivariable calculus. Peeking ahead at some differential geometry textbooks, I noticed a great deal of terminology, particularly with respect to sets, topological spaces, etc. So I'm trying to familiarize myself with that terminology.

Not sure what I need to focus on next after multivariable calculus. I've seen online textbooks involving algebraic topology, differential geometry, and others. I really want to work toward an understanding of space-time and general relativity.

**NateTG** · Mar11-04, 01:25 PM

I don't think that Differential Geometry requires a strong background in topology, but my experience with Differential Geometry hasn't exactly been the best, so someone else should probably comment on that.

http://math.ucr.edu/home/baez/

might help with your quest for information.

**bmegun** · Apr10-04, 07:06 PM

T is not a subset of X. T is a collection of subsets of X. In other words, T is a set of sets. Your first reply said this in a fancy way, by stating that T is a subset of the power set of X. To be a topology, T must contain at least two elements: X itself, and the empty set. If T contains only these two sets, it is called the indiscreet topology
for X. If T contains all subsets of X, i.e., the entire power set, then T is called the
discreet topology. Interesting topologies associated with X are usually between these two extremes.