Topology Definition and 818 Threads

Greetings all, doing more problems for my test tomorrow. I'm not sure how to start these two.. 1.) I'm trying to show that if X and Y are Hausdorff spaces, then the product space X x Y is also Hausdorff. So, I know that I must have distinct x1 and x2 \in X, with disjoint neighborhoods U1...

Here's two more question I'm working on in test prep. 2.) Let Y = [-1,1] have the standard topology. Which of the following sets are open in Y, and which are open in R. A= (1,1/2) \cup (-1/2,-1) B= (1,1/2] \cup [-1/2,-1) C= [1,1/2) \cup (-1/2,-1] D= [1,1/2] \cup [-1/2,-1] E= \cup...

Greetings all, I have a test upcoming next week and a lot of problems to solve. I asked in a previous thread whether or not I should just continue to adding to the thread and it seems the answer was yes. I appreciate any help! This is a tough subject. Anyways, the first one I'm working on. I...

Homework Statement Give an example of a separable Hausdorff space (X,T) that has a subspace (A,T_A) that is not separable. Homework Equations A separable space is one that has a countable dense subset. The Attempt at a Solution Let (X,T) (i.e. the separable Hausdorff space) be...

Homework Statement Let (X,T) be a topological space and let U denote the product topology on X x X. Let delta = {(x,y) in X x X : x = y} and let U_delta be the subspace topology on delta determined by U. Prove that (X,T) is homeomorphic to (delta,U_delta) The Attempt at a Solution...

Homework Statement Give an example of a separable Hausdorff space (X,T) with a subspace (A,T_A) that is not separable. The Attempt at a Solution well since a separable space is one that is either finite or has a one-to-one correspondence with the natural numbers, the separable...

Hello, I have a question about topology. If X is a path-connected space then is it also true that closure X is path-connected? I think it's obvious, but I can't solve it clearly...

Homework Statement Let Y := \prod_{i \in I} X_i Now assume U_i \subset X_i to be open. If we take i to be infinite, \prod_{i \in I} X_i cannot be open. Why? Homework Equations The Attempt at a Solution I can't quite get my head around how to approach this problem. A...

let X be a set and C be a collection of subsets of X whose union equal X. let β[SUB]C the collection of all subsets of X that can be expressed as an intersection of finitely many of the sets from C. let T[SUB]C be the topology generated by the basis β[SUB]C. prove that every set in C is...

Homework Statement Find a metric space and a closed, bounded subset in it which is not compact. Homework Equations N/A The Attempt at a Solution I know that a metric space is a space that has definition such that a point has a distance to any other point. However, I know a...

Homework Statement Let O be an open subset of R^n and suppose f: O --> R is continuous. Suppose that u is a point in O at which f(u) > 0. Prove that there exists an open ball B centered at u such that f(v) > 1/2*f(u) for all v in B. Homework Equations f continuous means that for any {uk} in O...

Homework Statement Let r be a positive number and define F = {u in R^n | ||u|| <= r}. Use the Componentwise Convergence Criterion to prove F is closed.Homework Equations The Componentwise Convergence Criterion states: If {uk} in F converges to c, then pi(uk) converges to pi(c). That is, the...

Hello, I have a general interest in teaching myself topology to build up to moving onto Representation theory. I have chosen M.A. Armstrongs's book "Basic Topology" as my start. My Question... where would you all recommend I go from there. I took top in undergrad and that was the book...

Determine the set of limit points of: A = { \frac{1}{m} + \frac{1}{n} \in R | m,n \in Z_{+} } I can see that everything less than one can't be reached by this set. Is my set of limit points (0,1) ?

Problem: Let P be a polygon with an even number of sides. Suppose that the sides are identified in pairs in accordance with any symbol whatsoever. Prove that the quotient space is a compact surface. Proof: Ok, here are some of my thoughts about the proof. I believe that one would need...

Is there anywhere in topology where one would see the Chinese Remainder Theorem?

Please if someone could help me understand something I saw in a proof. It's about proving that if X,Y is compact then their product (with product topology) is compact. Suppose that X and Y are compact. Let F be an open cover for XxY. Then, for y in Y, F is an open cover for Xx{y}, which is...

I have a Topology midterm tomorrow and I'm going through exercises in my book. Perhaps someone could let me know whether I ought to make a thread for each question or if I may continue adding to this thread... Determine which of the following collections of subsets of R are bases: a.) C1 =...

Hi. Can I have some help in answering the following questions? Thank you. Let {f_n} be a sequence of functions from N(set of natural numbers) to R(real nos.) where f_n (s)=1/n if 1<=s<=n f_n (s)=0 if s>n. Define f:N to R by f(s)=0 for every s>=1...

Homework Statement So, I'm going through a proof and it is shamelessly asserted that the collection of clopen sets of {0,1}^{\mathbb{N}} is a countable basis. Can anyone reasure me of this, point me in the direction of proving it. Thanks Tal

I debated whether to put this in this sub-forum or in the Topology & Geometry sub-forum, but I decided I'd give you guys the first crack at it: Take the union of all open intervals on the real numbers which do not include the number 1, call this union A. Then take the union of all closed...

Question is: X is a set and tau is a collection of subsets O of X such that X - O is either finite or all of X. Show this is a topology and completely described when a point x in X is a limit of a subset A in X. I already proved that this satisfies the conditions for defining a topology...

how does a lower limit topology strictly finer than a standard topology? please explain lemma 13.4 of munkres' topolgy..

i want to show that given any space X with the cofinite topology, the space X is sequentially compact. i have already shown that any space X with the cofinite topology is compact since any open cover has a finite subcover on X. i know that if we are dealing with metric spaces, then the...

Could anybody help me with this topology question? i) Prove that every map e: X-> R^n is homotopic to a constant map. ii) If f: X->S^n is a map that is not onto (surjective), show that f is homtopic to a constant map. It's part of a past exam paper but it does not come with solutions...

Consider the collection of sets C = {[a,b), | a<b, and and b are rational } a.) Show that C is a basis for a topology on R. b.) prove that the topology generated by C is not the standard topology on R.So, I know for C to be a basis, there must be some x \in R, and in the union of some C1...

Homework Statement Let (X,T) be a topological space, and let A be a subset of this space. Prove that there are at most 14 subsets of X that can be obtained from A by applying closures and complements successively. The Attempt at a Solution I understand the concept behind the theorem, that...

Homework Statement Let (X,T) be a topological space, let C be a closed subset of X, let U be an open subset of X. Prove that C - U is closed and U - C is open. The Attempt at a Solution I was trying to do this by 4 cases: Case 1: Let U be a proper subset of C. Then U - C = empty...

Recently a professor recommended Bott & Tu's Differential Forms in Algebraic Topology to me. My knowledge of algebraic topology is at the level of Munkres' book. Would Bott & Tu's book be too advanced for me to understand at this stage?

http://arxiv4.library.cornell.edu/abs/1101.3216 Fivebranes and Knots Edward Witten 146 pages (Submitted on 17 Jan 2011) Abstract We develop an approach to Khovanov homology of knots via gauge theory (previous physics-based approches involved other descriptions of the relevant spaces of BPS...

Hi all, I've been reading on phase changes that occur in manifolds such as flop transitions and conifold transitions for some time but I don't quite understand this one thing: flop transitions mathematically describe how one calabi-yau can change into another and most books mention that but...

Homework Statement One needs to show that a countable product of topologically complete spaces is topologically complete in the product topology. The Attempt at a Solution A space X is topologically complete if there exists a metric for the topology of X relative to which X is...

Hi, I have a question regarding the idea of eternal inflation happening in a multiverse and the topology of our universe. Looking at the current data it seems plausible that our universe is flat or slightly negatively curved, i.e. has a non-compact topology. But the idea of eternal...

Homework Statement Let (X,d) be a metric space and let A be a nonempty subset of X. Define a function f:X -> R^1 by f(x) = inf{d(x,a) : a is an element of A}. Prove that f is continuous. Homework Equations The Attempt at a Solution Intuitively I can see that the function is...

I need a suggestion for a name of a general topology book, I'd be grateful if there's a link for it.

Hello there! I just started reading Topological manifolds by John Lee and got one questions regarding the material. I am thankful for any advice or answer! The criteria for being a topological manifold is that the space is second countantable ( = there exists a countable neighborhood...

Homework Statement Find an inclusion map i from S^1 to RP^2 such that the induced map of the inclusion (by the fundamental group) is not the zero element. Known: pi_1(S^1) = Z and pi_1(RP^2) = Z/2Z Homework Equations Can we define i as a composite of two other inclusions? The...

Homework Statement For some reason, the uniform topology always causes me problems. So, let's work this through. Let Rω be given the uniform topology, i.e. the topology induced by the uniform metric, which is defined with d(x, y) = sup{min{|xi - yi|, 1}, i is in ω}. Given some n, let...

A follow-up of sorts on my https://www.physicsforums.com/showthread.php?t=457248". I've decided that, barring any technicalities that prevent me from getting a necessary override, I'm going to definitely take Topology next semester. (I need an override because Topology requires Linear Algebra...

Hey everybody. I'm a Pure Math major and I'm trying to finalize my schedule for next semester. Originally I was enrolled in a second ODEs course (focusing predominantly on systems of linear ODEs, existence and uniqueness theory, and qualitative solutions), but after a rough semester with PDEs...

Homework Statement Suppose A and B are disjoint closed sets in the metric space X and assume in addition that A is compact. Prove there exists ∆ > 0 such that for all a ∈ A, b ∈ B, d(a, b) ≥ ∆ 2. The attempt at a solution I really don't have an attempt at a solution because I am 100%...

Hi, I'm a junior undergrad majoring in math and physics, and am deciding between complex analysis and topology for next semester. (I'm planning on doing theoretical physics for grad, something on the more mathematical side, so topology would likely be used). Complex Analysis Pros...

Hi I would appreciate any help with these problems. Thanks in Advance! 1.Suppose X is the boundary of a manifold W, W is compact, and f : X → Y is a smooth map. Let w be a closed k-form on Y , where k = dimX. Prove that if f extends to all of W then integral of f* dw = 0. Assume that W...

Currently in my course in topology we have covered the point-set portion of the Munkres text, and the professor has moved into some additional material in which munkres has no resources, mainly the classification of surfaces. The professor let me borrow his resource for a while, but I was...

Homework Statement Let (X,Ʈ) be a topological space and T \subseteq X a compact subset. Show that T is compact as a subset of the space (T,Ʈ_T) where Ʈ_T is the relative topology on T. Homework Equations The Attempt at a Solution Hi everyone, Here's what I've done so far: T...

Homework Statement X={x | xn E R | 0\leq x \leq 1} d(x,y)= \Sigman=1infinity |xn - yn|*2-j Show: 1. (X,d) is a metric space 2. (X,d) is separable 3. (X,d) is compactHomework Equations n/aThe Attempt at a Solution Here we go. number 1. Show that d(x,y)=d(y,x): \Sigman=1infinity |xn - yn|*2-j =...

This is a qualifier exam question in algebraic topology: Let Z * Z_2 = <a, b | b^2> be represented by X = S^1 \vee RP^2 , i.e. the wedge of S^1 (the unit circle) and RP^2 (the real projective plane). For the subgroup H below construct the covering space ˜X by sketching a good picture for...

Does anyone know a very good introductory book to topology? I am looking for an introductory book with solved examples, proves and so on. Thanks

Hi, I'm more conversant with Physics than Biology, and I think this question may actually apply more to the computer sciences so pls bear with me - DNA is represented as a 'strand', and is analogous with a 'line' of code. Turing envisioned the computing process as two 'infinite' strings...

Hello I am curious about this. Uryshon' s lemma is also known as "the first non-trivial fact of point set topology", what are the others non-trivial facts of point set topology? I suppose Tychonoff' s theorem is another one.