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I am really confused about basic topology, What does it mean by E is closed if every limit point of E is a point of E.
and How to use the notation of neighborhood to prove things .
Suppose
where X is a metric space , how can E be relative to Y and not to X at the same time.Numbers are the essence of the Universe
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All right, lets start with basic definitions. Let X be a topological space. Then I would define a subset E of X as being closed in X iff the complement of E in X is an open set in X.
It turns out that E is closed in X if and only if E contains all its limit points in X. So what exactly is a limit point of E in X?
Let E be a subset of X. Then a point a ∈ X is said to be a limit point of E iff for every open set T of X such that a ∈ T, T∩E contains an element x such that x ≠ a. (Note that a does not have to belong to E.)
So let E be a closed set. We wanna show that it contains all its limit points. Let a be a limit point. Suppose a ∉ E. Then a ∈the complement E[sup]c[/sup]. So E[sup]c[/sup] is an open set containing a and so, by definition, E[sup]c[/sup]∩E should be nonempty. But it is empty. This contradiction means that it is impossible for a to not belong to E. ∴ E contains all its limit points.
The converse is probably more tricky. Ill be working on it and coming up with some answers shortly.
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Argh, its not that difficult after all.
Suppose E contains all its limit points. Let a belong to the complement of E in the topological space X. Then a is not a limit point of E and so there exists an open set
containing a such that . Hence E[sup]c[/sup] is a union of open sets and therefore, by the axiom of topological spaces, E[sup]c[/sup] is an open set. Hence, the complement of E[sup]c[/sup], namely E, is closed. QED.Last edited by JaneFairfax (2007-03-30 15:11:21)
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To answer the rest of your questions
How to use the notation of neighborhood to prove things .
I dont. Im not very clear on whats meant by neighbourhood myself.
Suppose
where X is a metric space , how can E be relative to Y and not to X at the same time.
Whats the definition of relative to with regards to metric spaces?
Last edited by JaneFairfax (2007-03-30 15:21:36)
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How to use the notation of neighborhood to prove things .
I dont. Im not very clear on whats meant by neighbourhood myself.
We can get a little insight if we look into the definition:
It's not to hard to see from here that an open set is a neighborhood of its points. So really in most cases when you are talking about open (sub)sets containing a point, such as has been done a few times in here, you are talking about a neighborhood of that point. A simple example of a neighborhood is an open ball. If you would like to see some proof explicitly using the term "neighborhood", just ask.
Suppose
where X is a metric space , how can E be relative to Y and not to X at the same time.Whats the definition of relative to with regards to metric spaces?
I think he might be "open relative to." In a metric space (M, d), the open ball of center a ∈ M and radius r is defined as B[sub]M[/sub](a, r) = {x ∈ M: d(x, a) < r}. Now if S is a subset of M, a point s ∈ S is called an interior point of S if some ball B[sub]M[/sub](s, r) lies entirely in S. As is the case in basic topology, S is called open if all of its points are interior points.
Now to answer Stanley's question, let X = R, Y = [-1, 1], and E = (0, 1], with each space having the standard Euclidean metric. 1 is an interior point of E relative to Y since B[sub]Y[/sub](1, 1) = (0, 1] lies entirely in E. Clearly all the other points of E are interior points relative to Y, so E is open relative to Y. But 1 is not an interior point of E relative to X, since B[sub]X[/sub](1, ε)
= (1 - ε, 1 + ε) will never be entirely contained in E with ε > 0. Then not all of the points of E are interior relative to X, and thus E is not open relative to X.
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Thanks a lots . But like this kind of expression metric space (M, d), What does it mean , I've never seen it before.
Last edited by Stanley_Marsh (2007-03-30 20:33:10)
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Did you guys feel confused at the first time? the whole thing is so abstract , my textbook doesnt have a graph and I have to learn it by myself .
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(M, d) denotes the metric space given by the set M equipped with the metric d. It's a shorthand method for specifying the set and the metric used in a metric space, and is pretty useful when you are considering two metric spaces with different metrics, such as (S, d[sub]1[/sub]) and (T, d[sub]2[/sub]). It is also easier than saying "let M be a set and d: M × M -> R be a metric in M."
Topology is certainly abstract, and it might be the most abstract mathematics you have come across yet. What is the title of your book and the name of the author(s)? I taught myself topology as well, and I must say everything is a lot easier when you don't do it as abstractly at first. A good way to learn basic topology is to do a lighter version which is applied to real analysis. Chapters 3 and 4 of Tom Apostol's "Mathematical Analysis" (2nd Edition) do this pretty well. If you could get access to this book you could get a better grip on the basic concepts, since Tom doesn't throw anything too complicated at you. The problems are very doable as well, but some of them can still provide a challenge. If you want an entire book containing plenty of introductory point set and algebraic topology that is easy enough of a read for self-teaching, consider "Topology" (2nd Edition) James Munkres. It might be the best general introduction to topology out there. Michael Henle's "A Combinatorial Introduction to Topology" is another good elementary book (it was my first ). It's a Dover book, so it'll only cost you a little more than $10 (my copy says $12.95) and you should be able to find it at a bookstore like Borders. "Topology" by Hocking and Young is another good Dover book (priced at $13.95 on my copy), but I must say that it is rougher than any of the others that I have mentioned, and it probably won't help you out nearly as much if you are reading it by yourself.
If you can't find any of these books at a local library and want to purchase them, I recommend using www.bookfinder.com. This website searches through most online bookstores and lists results by price in ascending order. In particular, I saved $90 on my copy of the Munkres text thanks to this website, so as you can see it is a very nice tool in helping you save money.
Last edited by Zhylliolom (2007-03-30 21:16:48)
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My book is Principles Of Mathematical Analysis by Walter Rudin , and I find his proofs very confusing . Thanks again!
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Ah, Rudin's text is very concise with everything. It is probably the kind of book that would be best used with a professor who can fill in the blanks for you. You'll have a hard time teaching yourself out of such a book. One thing I can recommend is that you try to prove the theorems in the chapter by yourself before reading his proof, since these are really the only thing you can check your work on in this book. Apostol's text ("Mathematical Analysis", as mentioned in my previous post) is a text at the same level as Rudin's but it is much easier to learn from. In addition, it has 5 more chapters than Rudin's book, so you get to read about more things such as functions of bounded variation, Fourier analysis (26 pages compared to Rudin's 7 pages on Fourier series), multiple Lebesgue integrals, complex analysis, etc. The only big topic that Rudin's text has that Apostol's doesn't is differential forms, but you can save that for differential geometry anyway. Apostol's text is nearly 200 pages longer as well, so you can tell that he gives some more explanations and examples in his work.
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Yah , I am going to try that one , thank you . and , I always wanted to learn Theoretical Physics someday , do u know what kind of math , I have to be equiped with?
Last edited by Stanley_Marsh (2007-03-31 11:40:26)
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It really depends on what exact theory you want to learn and how deep you want to get into it. Some of the math in particular areas of theoretical physics can be extremely nasty. A good background in linear algebra, vector and tensor analysis, abstract algebra, and differential geometry (probably topology and functional analysis too) is probably what you'll want to have some freedom in roaming around the world of modern theoretical physics. Are there any particular theories you are interested in?
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I am interested in finding the explanation for space time , black hole sth like that.
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It sounds like relativity and cosmology is your thing. You'll want to learn differential geometry first though. I don't really know any good beginner books on the subject though. Out of the 6 books on differential geometry that I have, they are either too elementary (first half of the book is like semi-advanced calculus, then they talk about multilinear algebra, then finally they talk about manifolds; I recommend a book that is on manifolds throughout) or they are too advanced (made for graduate courses in differential geometry, they assume you have a rigorous background in algebraic and point set topology, abstract algebra, and even category theory in one particularly advanced book). "A First Course in General Relativity" by Bernard F. Schutz is a pretty self-contained book on differential geometry and general relativity. As long as you know some electrodynamics, special relativity, and vector analysis you should be able to go through this book without much of a problem. My only problem with it is that in general I don't like learning math from a physics book, because it is often not very rigorous (definition-theorem-proof format) and therefore you only get a weak understanding of the mathematics behind the subject. But this book will teach you the rudimentary differential geometry you need for basic general relativity, and it has a chapter on black holes. A great book on general relativity is "Gravitation" by John Archibald Wheeler, Kip S. Thorne, and Charles W. Misner. This 1215 page monster is the bible of general relativity, and belongs in every physicist's collection. It's not in mine though because even a used copy costs $70 . But I will buy it one day. "Gravitation" is pretty advanced though, and I don't recommend it as a first text on GR.
Last edited by Zhylliolom (2007-03-31 14:49:09)
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I am looking for some thing about Differential Geometry now . I always learn some from here and some from there , never go deep into one.
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