Topology Definition and 798 Threads

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...

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...

Homework Statement I am asked to show that T=[particular point topology on R^2 ((0,0) being the particular point)] is equal to T'=[topology on R^2 from taking the product of R in the particular point topology (0 being particular point) with itself]. The Attempt at a Solution I'm...

Hi, I've read this article (http://www.space.com/scienceastronomy/mystery_monday_040524.html), which says: "In the new study, researchers examined primordial radiation imprinted on the cosmos. Among their conclusions is that it is less likely that there is some crazy cosmic "hall of mirrors"...

I'm totally stuck on these two. The first is... Let A be a subset of X; suppose r:X->A is a continuous map from X to A such that r(a)=a for each a e A. If a_0 e A, show that... r* : Pi_1(X,a_0) -> Pi_1(A,a_0) ...is surjective. Note: Pi_1 is the first homotopy group and r* is the...

Homework Statement What is the torus excluding a disc homeomorphic to? What is the boundary of a torus (excluding a disc)?The Attempt at a Solution RP^2 X RP^2? As a guess.

Good Morning, I am trying to prove that any 2 open intervals (a,b) and (c,d) are equivalent. Show that f(x) = ((d-c)/(b-a))*(x-a)+c is one-to-one and onto (c,d). a,b,c,d belong to the set of Real numbers with a<b and c<d. Let f: (a,b)->(c,d) be a linear function which i graphed to help me...

Homework Statement In topology, a f: X -> Y is continuous when U is open in Y implies that f^{-1}(U) is open in X Doesn't that mean that a continuous function must be surjective i.e. it must span all of Y since every point y in Y is in an open set and that open set must have a pre-image...

I need to show if the finite complement topology,T_3, and the topology having all sets (-inf,a) = {x|x<a} as basis ,T_5, are comparable. I've shown that T_3 is not strictly finer than T_5. But I'm not sure about other case. I need help.

Homework Statement I always get confused between countably many vs. uncountable. I suppose if one can index the points of a set , then it is countable. 1)So, anything that is finitie is countable. Anything that is infinite is also countable? Then what is uncountable, something that...

I am not a math student but I have read some basic stuff about topology just because it sounded interesting, and I was wondering if people could name some uses of it, because it does not seem to have very many. Also on a related question, how often are new branches of mathematics "invented"...

Many abstract mathematical concepts have their intuitive correspondences or geometrical meanings. such as differentiable is corresponding to "smooth", determinant is corresponding to "volumn",homolgy group is corresponding to "hole". 1.The question is whether "exact" and "exact sequence" have...

Homework Statement I saw the following statement in a proof that a second countable normal space is homeomorphic to the Hilbert cube: n = (B_i, B_j) where the = sign is replaced with an approximately equals which I do not know how to make in latex B_i and B_j are basis sets s.t. B_i...

Suppose I have some subset of R, not necessarily an interval, let it be denoted as A. I have some union (might be countable, might be finite, might be uncountable) of sets where each set is an open set of A and the union of the open sets is equal to A. Can I conclude that A is open? I am not...

[SOLVED]Finite-Compliment Topology and intersection of interior Homework Statement Given topological space (R^{1}, finite compliment topology), find counter example to show that Arbitary Intersection of (interior of subset of R^{1}) is not equal to Interior of (arbitary intersection of...

1) Prove rigorously that S={(x,y) | 1< x^2 + y^2 <4} in R^n is open using the following definition of an open set: A set S C R^n is "open" if for all x E S, there exists some r>0 s.t. all y E R^n satisfying |y-x|<r also belongs to S. [My attempt: Let x E S, r1 = 2 - |x|, r2= |x| - 1, r =...

I just discovered the following. But since half the things I find in topology turn out to be wrong, I feel I better check with you guys. What I convinced myself of this time is that if you have a function f:(X,T)-->(Y,S) btw topological spaces, and S' is a basis for S, then to show f is...

Dear all, I am interested in the connection between the smoothness of a planet and the gravitational acceleration at the surface. Specifically, what is the highest a mountain can be for different values of g? More pertitently, what % of the Earth's surface would be covered with water if the...

I'm trying to prove the following Theorem. Suppose T1 and T2 are topologies for X. The following are equivalent: 1. T1 is a subset of T2; 2. if F is closed in (X, T1), then F is closed in (X, T2); 3. if p is a limit point of A in (X, T2), then p is a limit point of A in (X, T1)...

This question pertains to physics, but has to do with the math. In order to find the flux of an electric field you can put a sphere around it and use that to find flux, since the amount leaving is the same at every point. My teacher said that if you put a cube around the field/charge, you would...

In Munkres' Topology he defines a Cartesian product AxB to be all (a,b) such that a is in A and b is in B. He says that this is a primative way of looking at things. And then defines it to be {{a},{a,b}} He says that if a = b then {a,b} will just be {a,a} = {a} and therefore will only be...