Topology Definition and 818 Threads

"Easiest" topology textbook/book I am having a terrible time learning topology. Abstract algebra comes easily, as does analysis but Topology is not making any sense whatsoever to me and I honostly try harder in it than my other classes and it gets me 1/10th the progress if not thousands less...

Homework Statement Let d be the usual metric on RxR and let p be the taxicab metric on RxR. Prove that the topology of d = the topology of p. Homework Equations The Attempt at a Solution I am trying to show that the open ball around point (x,y) with E/2 as the radius (in...

Hello all, I have a question I'm having a hard time with in an introductory Algebraic Topology course: Take two handlebodies of equal genus g in S^3 and identify their boundaries by the identity mapping. What is the fundamental group of the resulting space M? Now, I know you can glue two...

I've just started looking at Rudin (Real and complex analysis, the big one) in which he offers the following definition (1.2): a)A collection \tau of subsets of a set X is said to be a topology in X if \tau has the following three properties: i) \emptyset \in \tau and X \in \tau ii)...

As I understand it a topology on a set X is a collection of subset that satisfy three conditions 1) The collection contains X and the null set 2) It is closed under unions (perhaps a better way to say this is any union sets in this collection is again in the collection). 3) The intersection...

I am trying to show that if X is a topological space, ~ an equivalence relation on X and q:X-->X/~ the quotient map (i.e. q(x)=[x]), then the quotient topology on X/~ (U in X/~ open iff q^{-1}(U) open in X) is characterized by the following universal property: "If f:X-->Y is continuous and...

Hi...I'm new to the forum but I need help with the following question. I need to find a topology on N for which there are exactly k limit points. k is a positive integer. Tips I have received: find countable subsets in R...then a bijection will produce the needed topology on N? Any help is...

I understand that topo-physiologically, the human body is a donut. i.e. not only are we a donut physically, but we are a donut as an organism. Our skin is an interface between the outside world and the inside of our bodies. Bacteria and other microbes have to get through our skin defense...

I know the importance of Topology, but I need to know if not taking this course in my last year will make a big difference. I will be a senior math major and I may decide to go to graduate school. I will take a year off to try to find some work experience to decide exactly what I want to do...

i've texed up three proofs in from elementary topology. can someone please check them? actually i'll just retype them here for convenience 8.2.5 Let f: X_{\tau} \rightarrow Y_{\nu} be continuous and injective. Also let Y_{\nu} be Hausdorff. Prove : X_{\tau} is Hausdorff...

u is the usual topology, cf is the cofinite topology. yes proof: pick a and b in (R,cf) ((0,1),u) ~ ((a,b),u). then the identity on (a,b) is continuous is because (R,cf) \subset (R,u). map 0->a and 1->b. the fn is continuous at the end points because no subset of the image is open in...

Hi, does it make sense to posit a curved (hall of mirrors type) space without higher dimensions? In other words, if I say we live in a 3 dimensional torus shaped universe, does that statement necessarily entail there's at least a 4 dimensional overall hyperspace? Or can I have a torus curved...

So I'm trying to teach myself some topology, and the first thing I noticed, was that a metric space is a topological space under the topology of all open balls.. But then, consider the intersection of two open balls, can someone prove to me that the result is another open ball? Or do they...

I'm struggling with something that I suspect is very basic. How do I should that the closure of a connected set is connected? I think I need to somehow show that it is not disconnected, but that's where I'm stuck. Thanks

Hi, I have a conceptual question. In a project I am working on, we are dealing with \Re^{n} (with the usual topology), and I am working on characterizing some objects. In particular, I am dealing with the intersection of two simply connected open sets (that do not have any sort of...

[SOLVED] Topology of curved space Homework Statement The distance between a point (r, theta) and a nearby point (r + dr, theta + d\theta) on a positively curved sphere is given by ds^2 = dr^2 + R^2 \sin ^2 (r/R)d\theta ^2 NOTE: I mean that ds^2 = (ds^2). My question is - how do I...

I would just like to know which of these math courses is best suited for physics. I have taken advanced calculus and linear algebra, so I've seen most of the proofs one typically sees in an intro analysis course (ie. epsilon delta etc.). I intend to do work with a lot of Quantum Field Theory...

Hi, I have a question that I'm not sure about. If f:A->C is continuous and B is a subset of C that is simply connected, is f(^-1)(B) necessarily connected or simply connected for that matter? Since the spaces are not necessarily homeomorphic I cannot consider it a topological invariant...

I'm a physics student and I'm trying to work my way through Isham's Modern differential geometry for physicists. I guess the first question would be what you guys think of this book, does it cover all the necessary stuff (it's my preparation for general relativity)? Sadly I'm already having...

Homework Statement Let X be an ordered set. If Y is a proper subset of X that is convex in X, does it follow that Y is an interval or a ray in X? The Attempt at a Solution I considered it to be yes. Since in the ordinary situation, the assertion is obviously valid: check out the...

I can't seem to find this result in any of my textbooks. Given any basis B for a topology T on X, is there a minimal subset M of B that also is a basis for T (in the sense that any proper subset of M is not a basis for T)? If so, is Zorn's Lemma needed to prove this? Is the same true of...

I posted this earlier and thought I solved it using a certain definition, which now I think is wrong, so I'm posting this again: Show that the quotient spaces R^2, R^2/D^2, R^2/I, and R^2/A are homeomorphic where D^2 is the closed ball of radius 1, centered at the origin. I is the closed...

Show the following spaces are homeomorphic: \mathbb{R}^2, \mathbb{R}^2/I, \mathbb{R}^2/D^2. Note: D^2 is the closed ball of radius 1 centered at the origin. I is the closed interval [0,1] in \mathbb{R}. THEOREM: It is enough to find a surjective, continuous map f:X\rightarrow Y to show that...

Conflicting statements from topology textbooks Definitions: A point p is a limit point of A iff all open sets containing p intersects A-{p}. Let A' denote the set of all limit points of A. So far, so good. Cullen's topology book (1968) states that (A U B)' = A' U B'. I read her proof...

Let A, B in R^n be closed sets. Does A+B = {x+y| x in A and y in B} have to be closed? Here is what I've tried. Let x be in A^c and y in B^c which are both open since A & B are closed. So for each x in A^c there exists epsilon(a)>0 s.t. x in D(x, epsilon(a) is subset of A^c. For each y...

[SOLVED] Topology: Nested, Compact, Connected Sets 1. Assumptions: X is a Hausdorff space. {K_n} is a family of nested, compact, nonempty, connected sets. Two parts: Show the intersection of all K_n is nonempty and connected. That the intersection is nonempty: I modeled my proof after the...

I was wondering about topology. a) Is there an algorithm for the number of topologies on finite sets? b) If two spaces are homeomorphic, are intersections of opens sent to intersections of opens? Are unions of opens sent to unions of opens? I tried to find an algorithm in the first part, and...

[SOLVED] Hausdorffness of the product topology Is it me, or is the product of an infinite number of Hausdorff spaces never Hausdorff? Recall that the product topology on \Pi_{i\in I}X_i has for a basis the products of open sets \Pi_{i\in I}O_i where all but finitely many of those O_i are...

Homework Statement 1. given a set X and a collection of subsets S, prove there exists a smallest topology containing S 2. Prove, on R, the topology containing all intervals of the from [a,b) is a topology finer than the euclidean topology, and that the topologies containing the intervals of...

What is the biggest unsolved problem in topology

Let C(X,Y) be the continuous functions space between the topological spaces X,Y, with the open-compact topology. prove that if the sequence {f_n} of C(X,Y) converges to f0 in C(X,Y) then for every point x in X the sequence {f_n(x)} in Y converges to f0(x). here's what I did, let x be in X and...

[SOLVED] very basic topology questions Homework Statement Let X be a set and T be the collection of X and all finite subsets of X. When is T a topology? Let T' be the collection of X and all countable subsets of X, when is T' a topology? The Attempt at a Solution it's clear the empty set...

1. Let f:\mathbb{R}\rightarrow\mathbb{R} be a bijection. Prove that f is a homeomorphism iff f is a monotone function. I think I have it one way (if f is monotone, it is a homeomorphism), but I'm stuck on the other way (if f is a homeomorphism, then it is monotone). I tried to prove...

I have photocopied pages of a advanced-looking point-set topology textbook, but I don't know the name of the book or the author. It has 427 pages (the last index page is p.427), and based on its references, it was written no earlier than 1966, and probably no later than 1975. I've attached a...

I've started studying point-set topology a month ago and I'm hooked! I guess one reason is because each question is proof-based, abstract, and non-calculational, which is what I like. I've decided to take on the project of proving every single theorem in topology (that is found in textbooks)...

Hello I have a proof that I need to try to work out but I'm not really getting too far and need help if you could at all. The question is Let A and B be two subsets of a metric space X. Prove that: Int(A)\bigcupInt(B)\subseteqInt(A\bigcupB) and Int(A)\bigcapInt(B) = Int(A\bigcapB) I...

I'm having some trouble understanding the distinction between closed sets, open sets, and those which are neither when the set itself involves there not being a finite boundary. For example, the set { |z - 4| >= |z| : z is complex}. This turns out to be the inequality 2>= Re(z). On the right...

Does anyone here have one of these book readers? I want to know how good the 'experimental' PDF support is. For instance, will it display Alan Hatcher's Topology book?

I was just wondering if anyone had a decent website explaining some of the basic terminology of differential topology. Specifically, I'm having a bit of trouble understanding charts and atlases and how one defines a smooth manifold in an arbitrary setting (i.e., not necessarily embedded in R^n)

Let A\subset X be a subset of some topological space. If x\in\overline{A}\backslash A, does there exist a sequence x_n\in A so that x_n\to x? In fact I already believe, that such sequence does not exist in general, but I'm just making sure. Is there any standard counter examples? I haven't seen...

Homework Statement All right, so this appeared on my final. The intervals are in the reals: If f : [a, b] -> [c, d] , and the graph of f is closed, is f continuous? Homework Equations The Attempt at a Solution Well, my gut reaction is no, just because it seems like a fairly...

Not sure where to put a question about topology, but I'll try here. I'm trying to show that a certain topology for the Real line is not normal. The topology in question has no disjoint open sets (they are all nested) and therefore, no disjoint closed sets. If a topology has no disjoint...

just want to see if i got these: 1.let U be open in X and A closed in X then U-A is open in X and A-U is closed in X. 2. if A is closed in X and B is closed in Y then AxB is closed in XxY. my proof: 1. A'=X-A which is open in X X-(A-U)=Xn(A'U U)=A'U(U) but this is a union of open sets...

For less than BH_h, deep in gravitational potential well, with very extreme curvature, might one have a future light cone tipping over sufficiently to become spacelike and then wrap around to join up (glued) to past light cone? This is like a closed timelike curve, which can not be shrunk to a...

How useful is topology in theoretical physics? By topology, I mean the contents of Munkres book, Hausdorff spaces, homeomorphisms, etc. It seems to me like topology is totally a mathematical construct since the idea of an "open set" in an abstract space seems to have no "physical" meaning...

If topology of a manifold were invariant (or not), what specifically would topology of a patch of such manifold in neighborhood (outside) of such BH, suggest?

Well it's not homewrok cause i don't need to hand this question in, this is why i decided to put it here. (that, and there isn't a topology forum per se, perhaps it's suited to point set topology so the set theory forum may suit it). Now to the question: Show that if Y is a subspace of X...

Homework Statement If L is a straight line in the plane, describe the topology L inherits as a subspace of RlxR and as a subspace of RlxRl in each case it is a familiar topology.(Rl= lower limit topology) The Attempt at a Solution RlxR topology is the union of intervals...

I am looking for the most basic but rigorous to some extent book on Algebraic topology out there.

Note: I have many questions and will keep posting new ones as they come up. To find the questions simply scroll down to look for bold segments. Feel free to contribute any other comments relevant to the questions or the topic itself. Here it is... Let p:E->B be continuous and surjective...