What is the dimension theory?

1. Aug 24, 2014

Master Sashin

I have to do a project on the dimension theory buy i cant find any info on it. This is the wikipedia page and if you open it you can see its nonsense: http://en.wikipedia.org/wiki/Dimension_theory

so please could i get some info on the dimension theory (the mathenatical thoery)

AM i right in saying it deals with the fact there there are a infinite amount of dimensions?

2. Aug 24, 2014

micromass

Not at all. Dimension theory tries to associate a dimension with every topological space. For example, the topological space $\mathbb{R}^n$ should have dimension $n$.

There are various concepts which fit the bill, like the inductive dimensions.

As a start, try to read Munkres' topology book, chapter 8.

3. Aug 24, 2014

Master Sashin

Please explain that statement to me

4. Aug 24, 2014

micromass

Are you familiar with a topological space? Are you familiar with linear algebra and dimension there?

5. Aug 24, 2014

Master Sashin

SOrt of

6. Aug 24, 2014

micromass

Then I don't really see how my post is unclear. Can you explain.

7. Aug 24, 2014

Master Sashin

Im 15 maybe explain in a bit more

8. Aug 24, 2014

Master Sashin

wait are you saying that any real number raised to the power of another number such as "n" would have a dimension of "n". SO if 3 is raised to the power of 3 then it means it has 3 dimensions

9. Aug 24, 2014

Jorriss

What class is this for?

What do you actually know about linear algebra and topology? What does "sort of" capture?

10. Aug 24, 2014

Fredrik

Staff Emeritus
No, he's saying that the vector space of ordered n-tuples of real numbers is an n-dimensional vector space. For example the vector space of ordered triples of real numbers is a 3-dimensional vector space.

The question you just asked shows that you don't know what linear algebra is. That's OK, but we need to know these things to know how to answer your questions. So can you please tell us more about what you're studying and what sort of project you're supposed to do?

11. Aug 24, 2014

Master Sashin

Well its just explaining the dimension theory. A presintation explaining the dimension theory. I know that linear algebra is any eqn that has is a straight line on a graph. I know that topological means geometry...I know my knowlage is bad. What does vector mean, and tuples

12. Aug 24, 2014

Master Sashin

can you please just explain the fundemental laws or principles of the theory (im 15 so dont use too complicated terms)

13. Aug 24, 2014

Jorriss

There's a good chance what you mean by dimension theory is not what we mean by dimension theory. There is no way to explain dimension theory in a way that will make sense.

What is this for exactly?

14. Aug 24, 2014

micromass

I'm sorry, but I think that is not possible. You need quite some prerequisites in order to understand even the basics of the theory, and these are prerequisites that you clearly don't have yet (don't worry, I didn't have them either when I was 15).

The thing is that dimension theory requires topology as basic language which requires familiarity with basic analysis. Much of the motivation of dimension theory comes from linear algebra, since that is where you meet the concept of dimension for the first time. At its core, dimension is a number that tells you the degrees of freedom a space has. For example, you need 2 numbers to describe a point in the plane, so the plane is two-dimensional. This notion has been extended to a lot of different geometrical contexts. For example, in topology something is n-dimensional if its boundaries are n-1 - dimensional.

But I'm afraid you will have quite some study to do in order to understand dimension theory. If you are already acquainted with basic geometry such as dot products, then you can start with learning linear algebra right now. Good books are Lang's book "introduction to linear algebra" and Meyer's "Matrix Analysis and Applied Linear Algebra".

15. Aug 24, 2014

micromass

16. Aug 24, 2014

Fredrik

Staff Emeritus
I'm kind of bored at the moment, so I'm going to explain the basics. An ordered pair of real numbers is written as (x,y). It's "ordered" in the sense that (x,y) is usually not the same as (y,x). For example, (3,2) isn't equal to (2,3). Two ordered pairs (a,b) and (c,d) are equal if and only if a=c and b=d.

The sum of two ordered pairs is defined by this formula: (a,b)+(c,d)=(a+c,b+d)
The product of a real number and an ordered pair of real numbers is defined by this formula: a(b,c)=(ab,ac).

In this context, the ordered pairs are called "vectors" and the numbers are called "scalars". The operations defined above are called "addition" and "scalar multiplication"

It's common to write vectors in bold and real numbers not in bold. For example, $\mathbf x=(x_1,x_2)$. For a person who has some experience with proofs, it's not hard to show that these ordered pairs, and the operations of addition and scalar multiplication defined above, satisfy a number of conditions.

1. For all $\mathbf x, \mathbf y, \mathbf z$, we have $\mathbf x+(\mathbf y+\mathbf z)=(\mathbf x+\mathbf y)+\mathbf z$.
2. For all $\mathbf x, \mathbf y$, we have $\mathbf x+\mathbf y=\mathbf y+\mathbf x$.
...

There are 8 of these conditions. The full list can be found here: http://en.wikipedia.org/wiki/Vector_space#Definition

Ordered pairs are not the only thing that satisfies these conditions. Consider e.g. functions that take real numbers to real numbers. We can define the sum of two functions by saying that f+g is the function such that (f+g)(x)=f(x)+g(x) for all real numbers x. We can define the product of a number k and a function f by saying that kf is the function defined by (kf)(x)=k(f(x)) for all x. Then we can prove that these two operations satisfy the same list of conditions as the two operations we defined earlier:

1. For all f,g,h, we have f+(g+h)=(f+g)+h.
2. For all f,g, we have f+g=g+h.
...

The "vector space" concept generalizes this idea by saying that if V is any set (like the set of ordered pairs of real numbers, or the set of functions that take real numbers to real numbers) with two operations that satisfy these conditions, then we call that set a "vector space". The elements of that set are then called "vectors". So the two examples above are actually two examples of vector spaces.

The dimension of a vector space can be defined in several different ways. I like to define "linearly independent" first, and then define the dimension like this: A vector space V is said to be n-dimensional if it contains a linearly independent set with n vectors, but none with n+1 vectors. If V is n-dimensional, we call n the dimension of V.

Unfortunately the concept of linear independence is a bit tricky and takes a while to understand. I won't try to explain it here. You will have to look it up.

If a vector space contains a linearly independent set with n vectors, for all positive integers n, then that vector space is said to be infinite-dimensional. A vector space that isn't infinite-dimensional is said to be finite-dimensional. Linear algebra is (roughly) the study of finite-dimensional vector spaces. It doesn't have a lot to do with lines.

I have mentioned two different vectors spaces in this post. The first one is 2-dimensional. The second is infinite-dimensional.

Last edited: Aug 24, 2014
17. Aug 24, 2014

Fredrik

Staff Emeritus
Guys, there's no way that he meant "dimension theory" in the sense of topology. I don't even know what that is. He doesn't even know what linear algebra is, so it's almost certain that he's just talking about the possibility to define vector spaces that are more than 2-dimensional.

18. Aug 24, 2014

micromass

I know there's no way he meant it, but he did link to the wiki page where they discuss dimension theory in topology. So the OP should explain what he is asking this for and how he found that wiki page.

19. Aug 25, 2014

Master Sashin

Hey fredrik thanks alot...Im sure i meant the one about topology. Im in grade ten and im good at math but i dont know the terms. Give me a question on linear algebra and i should be able to answer it... and topolgy just so i know i know how to answer it

20. Aug 25, 2014

Jorriss

You didn't know what $\mathbb{R}^n$ meant. You don't know linear algebra.

Anyhow, you may want to pick another topic if you have a choice. This is too advanced of a topic.

Though it would be nice if you would say what the assignment is already.

21. Aug 25, 2014

micromass

Right, you're good at math but perhaps don't know the terms. And that is problematic because studying dimension theory will require you to know these terms first. So you first need to go through a linear algebra and topology book to get familiar with the terms.

1) Linear algebra: Determine the dimension and a basis for the kernel of the linear map associated with the matrix

$$\left(\begin{array}{ccc} 1 & 2 & 3\\ 4 & 5 & 6\end{array}\right)$$

2) Topology: Consider a uncountable set $X$ with topology
$$\mathcal{T}= \{A\subseteq X~\vert~X\setminus A~\text{is finite}\}\cup \{\emptyset\}$$
Determine whether this is a compact topological space and determine whether it is Hausdorff.

These are two very standard questions on linear algebra and topology.

22. Aug 26, 2014

Stephen Tashi

There is a topic in physics called "dimensional analyisis" that deals with deducing things about equations describing phyiscal situations from the fact that the physical units must "work out" correctly. http://en.wikipedia.org/wiki/Dimensional_analysis

23. Aug 26, 2014

Fredrik

Staff Emeritus
I don't think your assignment has anything to do with topology. Grade 10 teachers don't know topology, and even if your teacher is an exception, he/she wouldn't give an assignment like this to a student except maybe to mess with you. It would be like telling you that you have to write an essay on Vietnamese poetry in Portugese (assuming that you are neither Vietnamese nor Portugese). You would have to learn two new languages.

Each of the topics linear algebra, (university-level) calculus and topology includes more information than all high school math put together. It would take you at least a year of full-time studies to learn them adequately.

Can you please tell us how this assignment was given to you, and how you came to the conclusion that it's about the dimension concept in topology?

Last edited: Aug 26, 2014
24. Aug 26, 2014

mathwonk

the classic text is "dimension theory" by hurewicz and wallman, take a look.