Question on Topological Spaces

  • Thread starter scott
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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

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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

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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 [tex]\Re^n[/tex], for some n. The ordered n-tupes in [tex]\Re^n[/tex] 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 [tex]\Re^n[/tex] is [tex]d(\vec{x},\vec{y})=|(\vec{x}-\vec{y})|[/tex] where the vertical bars represent the vector norm.

My appologies regarding the gibberish comment.
 

NateTG

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Metric spaces are a type of topology, where all open sets [tex]O[/tex] have the property that [tex]\forall x \in O \exists \delta s.t. d(x,y) < \delta \rightarrow y \in 0[/tex]

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

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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.
 
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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.
 

chroot

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Just learn differential geometry and tensor calculus, and you'll be set for a good into to GR. There are many good books on the topic; I like one called "A Short Course on General Relativity" by Nightingale.

- Warren
 

mathwonk

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this discussion persuades me that the art of reading books is hopelessly lost. anyone who read even a few pages of a mediocre book on topology would not have any of these questions.

web based learning is obviously a satanic invention of the powers of ignorance, superstition, and darkness!
 

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