Topology Definition and 816 Threads

So I'm planning to delve into both of these subjects in some depth during the summer to prepare for undergrad analysis (using rudin) and a graduate differential topology class. My question is which one should I start out with and pay more attention to. I obviously need to study a lot of topology...

Hello all. I have to present a proof to our Intro to Topology class and I just wanted to make sure I did it right (before I look like a fool up there). Proposition Let c be in ℝ such that c≠0. Prove that if {an} converges to a in the standard topology, denoted by τs, then {can}...

Homework Statement Cl(S \cup T)= Cl(S) \cup Cl(T)Homework Equations I'm using the fact that the closure of a set is equal to itself union its limit points.The Attempt at a Solution I am just having trouble with showing Cl(S \cup T) \subset Cl(S) \cup Cl(T). I can prove this one way, but I...

i have one simple question if we a consider subsets of R^2 which are: a finite set and set of all integers, then aren't a finite set and set of all integers not closed? For instance for set of all integers, it do not have any limit points. thus by definition of closed (E is closed if all...

My brain is giving me confusions. Which of these is true? 1) Given a topology T and basis B, a set U is open iff for every x in U there exists basis element B with x belonging to B, and B contained in U. 2) Given a topology T and basis B, a set U is open iff for every x in U there exists open...

Got it, thank you

Hello everyone. I want to study topology ahead of time (it begins next semester only) and I have two options: I could go straight for general topology (among the books I searched I found Munkres to be the one I felt most comfortable with) or go for a thorough study of metric spaces (in which...

I am told that the interval (a, ∞) where a \in (0, ∞) together the empty set and [0, ∞) form a topology on [0, ∞). But I thought in a topology that the intersection if any two sets had to also be in the topology, but the intersection of say (a, ∞) with (b, ∞) where a<b is surely (a,b) which...

I would like to study the path components (isotopy classes) of the diffeomorphism group of some compact Riemann surface. To make sense of path connectedness, I require a notion of continuity; hence, I require a notion of an open set of diffeomorphisms. What sort of topology should I put on the...

Hi! I am a self-learner. What background knowledge is necessary to learn Analysis and Topology?

If τ is the co-finite topology on an infinite set X, does there exist an injection from τ to X? I'm having trouble wrapping my mind around this. on the one hand, for A in τ, we have A = X - S, for some finite set S. so it seems that there is a 1-1 correspondence: A <--> S, of τ with the...

[topology] "The metric topology is the coarsest that makes the metric continuous" Homework Statement Let (X,d) be a metric space. Show that the topology on X induced by the metric d is the coarsest topology on X such that d: X \times X \to \mathbb R is continuous (for the product topology on X...

Hello, A bit of background, I intend to major in physics and mathematics, and I am currently in second year. As it stands at the moment I am only enrolled in three units, and I was wondering If I should do, normally a third year unit, Introduction to Geometric Topology, (i can apply for an...

Homework Statement It might not be a real topology question, but it's an exercise question in the topology course I'm taking. The question is not too hard, but I'm mainly doubting about the terminology: Homework Equations N.A. The Attempt at a Solution I would think not, unless I'm...

Topology Proof: AcBcX, B closed --> A'cB' Homework Statement Prove: AcBcX, B closed --> A'cB' and where the prime denotes the set of limit points in that set X\B is the set difference Homework Equations Theorem: B is closed <--> For all b in X\B, there exists a neighborhood U...

I've came across a book about topolgy, Topology Without Tears by Sidney A. Morris. It can be found here: http://uob-community.ballarat.edu.au/~smorris/topology.htm The explanations are rather clear and an outline of the proof is given before each proof. However, many quite important concepts...

Homework Statement X - topological compact space \Delta = \{(x, y) \in X \times X: x=y \} \subset X \times X \Delta = \bigcap_{n=1}^{\infty} G_{n}, where G_{1}, G_{2}, ... \subset X \times X are open subsets. Show that the topology of X has a countable base. Homework Equations The Attempt...

Homework Statement Let Tx and Ty be topologies on X and Y, respectively. Is T = { A × B : A\inTx, B\inTy } a topology on X × Y? The attempt at a solution I know that in order to prove T is a topology on X × Y I need to prove: i. (∅, ∅)\inT and (X × Y)\inT ii. T is closed under...

Just to make sure that I'm not overlooking anything, is the following an example of a quotient map p: X \to Y with the properties that Y is pathwise connected (i.e. connected by a continuous function from the unit interval), \forall y \in Y: p^{-1}(\{ y \}) \subset X also pathwise connected and...

Homework Statement Let X be a compact and locally connected topological space. Prove that by identifying a finite number of points of X, one gets a topological space Y that is connected for the quotient topology. Homework Equations The components of a locally connected space are open...

Hello i studied Sadri Hassani az mathematical physics book. if i want to learn topology (( for general relativity )) what it the best book for introduction ?

I'm pursuing dual degrees in mathematics and computer science with a concentration in scientific computing and am trying to decide whether I should take intro to topology or number theory. Interests in no order are computational complexity, P=NP?, physics engines, graphics engines...

Topology of charm decays. Help!:) Hello Everyone:) It's my first post ever and I'm asking for help, sorry! I have to know how to identify charm decays in the films of the Na27 experiments, done in the 80's. It was used a bubble chamber and a spectrometer... In the paper it's said that...

Hello, Just out of curiosity, where would following "seperation axiom" fit in? So far I'm only acquainted with the T1, T2, T3 and T4 axioms (and the notion of completely regular in relation to the Urysohn theorem).

Hey guys, I want to study algebraic topology on my own. I just finished a semester of pointset topology and three weeks of algebraic topology. We did not use a textbook. Can anyone recommend a book on algebraic topology? Hatcher is fine but it is not as rigorous as I want. Munkres has...

Homework Statement I've been given the following problem: "Suppose that U is a finite-dimensional subspace of a Fréchet space (V,\tau). Show that the subspace topology on U is the usual topology (given for example by a Euclidean norm) and that U is a closed linear subspace of V." I feel a bit...

Hi! I have this two related questions: (1) I was thinking that \mathbb{Q} as a subset of \mathbb{R} is a closed set (all its points are boundary points). But when I think of \mathbb{Q} not like a subset, but like a topological space (with the inherited subspace topology), are all it's...

I have a friend who, like me, is a Math major, although she started later than I did and as such, hasn't yet gotten into the core classes for her degree. She's frequently checked out my own personal library and I figured that, since the holidays are coming up, it might be cool to start her off...

Does anyone know what \Lambda means in the collection of elements:\{ A_{\lambda} : \lambda \in \Lambda\} For example in the definition of a topology: if \{ A_{\lambda} : \lambda \in \Lambda\} is a collection of elements of a topology then \bigcup _{\lambda \in \Lambda} A_{\lambda} is in the...

Hi. I wanted to know in what way the group of translations on a real line with discrete topology (let's call it Td) will be different from the group of translations on a real line with the usual topology (lets call it Tu)? Is Td a Lie Group? Will it have the same generator as Tu?

Hi! I'm a beginner for a subject "topology". While studying it, I found a confusing concept. It makes me crazy.. I try to explain about it to you. For a set X, I've learned that a metric space is defined as a pair (X,d) where d is a distance function. I've also learned that for a set...

Homework Statement I need two counter examples, that show the following two theorems [B]don't/B] hold: Let X be a topological space. 1. If from the closeness of any subset A in X follows compactness of A, then X is compact. 2. If from the compactness of a subset A in X follows closeness...

Hi, I am trying to decide whether I should take a modern algebra or topology course next semester. I have a bachelor's in physics but I have not taken very many higher math classes. This is a list of the relevant classes I have taken. Calculus (up through partial differential equations)...

Hi. I did my undergraduate work in mechanical engineering and I am working on a PhD in fluid mechanics right now. I am interested in expanding my mathematical toolbox to include topology and am looking for some advice on where to start. What subjects/topics should I cover as a prerequisite to...

Hi, I want to study differential topology by myself, and i am looking for a clear book that emphesizes also the intuitive aspect. I will be grateful to get some recommendations. Thank's Hedi

I must say thusfar I read through chapter one of May's book and chapter 0 of Hatcher's, May is much more clear than Hatcher, I don't understand how people can recommend Hatcher's text. May is precise with his definitions, and Hatcher's writes in illustrative manner which is not mathematical...

Hi, the problem I am referencing is section 33 problem 4. Let X be normal. There exists a continuous function f: X -> [0,1] such that fx=0 for x in A and fx >0 for x not in A, if and only if A is a closed G(delta) set in X. My question is about the <= direction. So let B be the...

Homework Statement E is a compact set, F is a closed set. Prove that intersection of E and F is compact Homework Equations The Attempt at a Solution On Hausdoff space (the most general space I can work this out), compact set is closed. So E is closed. So intersection of E and F is...

Homework Statement The true problem is too complicated to present here, but hopefully somebody can give me a hand with this simplified version. Consider the set H = \{ (x,y) \in \mathbb R^2 : y \geq 0 \} . Denote by \partial H = \{ (x,0) \}. Let U and V be open sets (relative to H) such that...

Hi, I am enrolled in an Msc programme in pure maths, I wanted to ask for your recommendations on taking a basic graduate course in Algebraic Topology. Basically my interest spans on stuff that is somehow related to analysis, geometry or analytic number theory. The pros for choosing this...

Hi all! I haven't posted here in some time, and I am in need of the expertise of you fine folks. I am busy doing some work on spin geometry. Now, as you guys know, spin structures exist on manifolds if their second Stiefel-Whitney class vanishes. This class is an element of the second...

Not sure if this is the correct place to post this, please move if need be. I am currently learning about limit points in my Topology class and am a bit confused. Going by this: As another example, let X = {a,b,c,d,e} with topology T = {empty set, {a}, {c,d}, {a,c,d}, {b,c,d,e}, X}. Let...

[Topology] why the words "finer" and "coarser"? Hello, I'm following an introductory course on topology. Why is it that a topology with lots of opens is called fine, and one with a few ones is called coarse? More specifically: why is this terminology more logical than the reverse (i.e...

Trying to prove: The usual topology is the smallest topology for R containing Tl and Tu. NOTE: for e>0 The usual topology: TR(R)={A<R|a in A =>(a-e,a+e)<A} The lower topology: Tl(R)={A<R|a in A =>(-∞ ,a+e)<A} The upper topology: Tu(R)={A<R|a in A =>(a-e, ∞)<A} 3. The Attempt at a...

If a point particle is a string viewed end on in three dimensions, or is a zero brane, isn't this merely a perspective? Are open strings really more like ribbons given the dimension of time and are closed strings more like tubes seen end on? If so, could this mean that the string's length is...

Hi, All: I am trying to understand better the similarity between the compactness theorem in logic--every first-order sentence is satisfiable (has a model) iff every finite subset of sentences is satisfiable, and the property of compactness : a topological space X is said...

I am a Mathematics and Physics double major, currently in my second year. I really enjoy both subjects, but my interests are progressing towards Theoretical physics/mathematical physics. My academic goal is to improve my understanding of how the universe works and thus I would like to pursue a...

I'm making this thread because in a few weeks I'll be starting with teaching a topology course. I think I did pretty well last time I teached it, but I want to do some new things. The problem with the system in our country is that we hardly assign problems that students should solve. Students...

Can I simply combine the unions into a single interval like I did in (a)? The closed interval [1,4] fills in the (3) hole from (0,3), etc. I did something similar in (b). http://i111.photobucket.com/albums/n149/camarolt4z28/IMG_20110906_221620.jpg...

I know there are some threads out there already, but none really help me (see my description below). I am a high school student. My highest level of math education is Calculus I. I am currently taking Calculus II (although I already know the integration portion of this course). I have no...