Topology Definition and 818 Threads

Homework Statement Let (X_a, \tau_a), a \in A be topological spaces, and let \displaystyle X = \prod_{a \in A} X_a. Homework Equations 1. Prove that the projection maps p_a : X \to X_a are open maps. 2. Let S_a \subseteq X_a and let \displaystyle S = \prod_{a \in A} S_a \subseteq...

Homework Statement Let (X,d) be a metric space, M a positive number, and f: X->X a continuous function for which: d(f(x), f(y)) is less than or equal to Md(x,y) for all x, y in X. Prove that f is continuous. Use this to conclude that every contractive function is continuous. The...

Having some knowledge of differential geometry, I want to self-study topology. Which of the two areas shall I study first? Thanks for answer!

Homework Statement show that T:=(A subset X |A = 0 or X\A is finite) is a topology on X, Homework Equations We need to show 3 conditions. 1: X,0 are in T 2: The union of infinite open set are in T 3: The finite intersections of open sets are open. The Attempt at a...

Hi Guys I was wondering if anyone knows of a good link that shows why the finite complement is a Topology? I been told it is the finest topology is this right?

Homework Statement Find two spaces A,B compact where A and B are nonhomeomorphic but AX[0,1]\congBX[0,1] Homework Equations Definitions of homeomorphism, cardinality possiby, I have no idea where to start. The Attempt at a Solution My idea Is [0,1] and S^1, but I am not sure if the...

Homework Statement Verify that taking \mathbb{R}, the empty set and finite sets to be closed gives a topology. Homework Equations The Attempt at a Solution Clearly the empty set is finite as it has 0 elemnts, and so is closed. If X_i , for i= {1,...,n}, are finite sets then...

Homework Statement verify that R, the reals, quotiented by the equivalence relation x~x+1 is S^1 Homework Equations The Attempt at a Solution All i can think of is to draw a unit square and identify sides like the torus, but this would be using IxI, a subset of R^2, and gives a...

Let (X,d) be a metric space. Show that if there exists a metric d' on X/~ such that d(x,y) = d'([x],[y]) for all x,y in X then ~ is the identity equivalence relation, with x~y if and only if x=y. i have: assume x=y then d(x,y)=0 and [x]=[y] which implies d'([x],[y])=0 also. now...

Let \mathbb{RP}^n= ( \mathbb{R}^{n+1} - \{ 0 \} ) / \sim where x \sim y if y=\lambda x, \lambda \neq 0 \in \mathbb{R} adn the equivalence class of x is denoted [x]. what is the necessary and sufficient condition on the linear map f : \mathbb{R}^{n+1} \rightarrow \mathbb{R}^{m+1} for the...

Homework Statement Let X be a set equipped with a topology tau1, and let tau2 be the cocountable topology in which a set V in X is an open set if V is empty or X - V has only finitely or countably many elements. Consider the topology tau consisting of all sets W in X such that for each point p...

Homework Statement I have this schematic for a clapp oscillator (see "ClappwBias") which includes the DC biasing network. When broken up into feedback topology, I end up with something like "betanet", where the output is taken from the source of the MOSFET. Homework Equations...

I am a senior in mathematics studying graduate point-set topoology atm. I am thinking I want to study differential topology in graduate school and maybe apply it to problems in cosmology. Do I need to take more ODE and PDE? I took intro to diff eq- the one that all engineering undergrads take...

Is calculus enough?

How to compare the topology on R generated by the subbasis S={[x,y)|x,y are rational}U{(x,y]|x,y rational} to the discrete topology on R?

different metrics... same topology? Let (X,d) be a metric space. Define d':X x X-->[0,infinity) by ... d(x,y) if d(x,y)=<1 d'(x,y)= ... 1 if d(x,y)>=1 Prove that d' is a metric a d that d a d d' define the same topology on X. This is a weird seeming metric. I am not sure...

Let A, B be two connected subsets of a topological space X such that A intersects the closure of B . Prove that A ∪ B is connected. I can prove that the union of A and the closure of B is connected, but I don't know what to do next. Could anyone give me some hints or is there another way to...

Homework Statement Find three disjoint open sets in the real line that have the same nonempty boundary. Homework Equations Connectedness on open intervals of \mathbb{R}. The Attempt at a Solution If this is it all possible, the closest thing I could come up with to 3 disjoint...

I have a question regarding the Coulomb interaction in spaces with non-trivial topology. Suppose we have D large spatial dimensions (D>2). Then the Coulomb potential is VC(r) ~ 1/rD-2. Usually one shows in three dimensions that the Coulomb potential VC(r) is nothing else but the Fourier...

How do you define the inside and the outside of a loop drawn on a closed surface? For example, take a sphere. Draw a small circle around the point that is the North pole. Now you can expand the circle by pulling it down and stretching it until it fits around the equator. If you pull it down...

Hello, Suppose that I have a cell complex and I want to define it's geometric realization, I can do it via mapping such that assign coordinates to 0-cells. however how can i do that for edges, faces and volumes. is there is ageneral formulas for lines, faces and volumes. Regards

I'm having trouble understanding the definition of a Subspace/Induced/Relative Topology. The definitions I'm finding either don't define symbols well (at all). If I understand correctly the definition is: Given: -topological space (A,\tau) -\tau={0,A,u1,u2,...un} -subset B\subsetA...

Hey guys, I'm reading Munkres book (2nd edition) and am caught on a problem out of Ch. 2. The problem states: If {Ta} is a family of topologies on X, show that (intersection)Ta is a topology on X. Is UTa a topology on X? Sorry for crappy notation; I don't know my way around the symbols...

Homework Statement Given S1={(x,y) in R2: x2+y2=1}. Show that S1 is a 1-dimensional manifold. Homework Equations The Attempt at a Solution Let f1:(-1,1)->S1 s.t. f1(x)=(x,(1-x2)1/2). This mapping is a diffeomorphism from (-1,1) onto the top half of the circle S1. I was...

Homework Statement Let A denote a 3x3 matrix with positive real entries. Show that A has a positive real Eigenvalue. Homework Equations This is a problem from a topology course, assigned in the chapter on fundamental groups and the Brouwer fixed point theorem.The Attempt at a Solution I...

I won't be able to take a course in topology before I have to take the mathematics subject test of the GRE. I have the Princeton Review guide, but I'm looking for a little stronger of a foundation. Is there a good book introductory book for this purpose? If so, what sections should I read?

I'm self studying topology and so I don't have much direction, however I found this wonderful little pdf called topology without tears. So to get to the meat of the question, given that \tau is a topology on the set X giving (\tau,X), the members of \tau are called open sets. Up to that point...

[Topology] Product Spaces :( Homework Statement 1. Show that in the product space N^N where the topology on N is discrete, the set of near-constant functions is dense (near constant function is a function that becomes constant from a specific index..)... 2. Prove that in R^I the set of...

Hi all, Like the title suggests, i am interested in finding a topic in topology that would serve as the basis for a research paper. Since i am currently taking a first course in Topology (Munkres), i am basically looking for something that is not too advanced. So far i haven't been able to...

I have a question here and I'm not sure what to do as it always confuses me, any help? Let A,B be closed non-empty subsets of a topological space X with AuB and AnB connected. (i) Prove that A and B are connected. (ii) Construct disjoint non-empty disconnected subspaces A,B c R such...

Homework Statement Let A, B be closed non-empty subsets of a topological space X with A \cup B and A \cap B connected. Prove that A and B are connected. Homework Equations A set Q is not connected (disconnected) if it is expressible as a disjoint union of open sets, Q = S...

I found it is hard for myself to follow the book on general topology by willard, since there are too many abstract definitions with too few examples to help me to establish these terms. I am wondering if there is any good problem book with sufficient problems that would help to make abstract...

Hi All, In my control systems lectures my professor talks about feedback changing the 'topology' of the system. Is he just talking about the structure of the block diagram changing, or is there some link to the topology which mathematicians refer to Regards, Thrillhouse86

I see that there are four different GTM textbooks on the subject. Which one of these is the most suitable for self-study? GTM 56: Algebraic Topology: An Introduction / Massey GTM 127: A Basic Course in Algebraic Topology / Massey GTM 153: Algebraic Topology / Fulton I want to pick up...

Hi everyone, I am stuck with 2 problems from Munkres' book and I would appreciate if someone helped me solve them. Thank you in advance. Here they are: 1. Consider the sequence of continuous functions fn : ℝ -> ℝ defined by fn(x) = x/n . In which of the following three topologies does...

Hey guys, i am studying cohomology by hatcher's. Could anyone provide me some ideas on these problems? Thank you all! Let f : S2n-1 -> Sn denote a continuous map. Let Xf = D2n union f Sn be the space obtained by attaching a 2n-dim cell to Sn using the map f. i). Calculate the integral...

Hey guys, i am studying cohomology by hatcher's. Could anyone provide me some ideas on these problems? Thank you! Let f : S2n-1 -> Sn denote a continuous map. Let Xf = D2n union f Sn be the space obtained by attaching a 2n-dim cell to Sn using the map f. i). Calculate the integral...

Is there any formula that gives a relation between the cardinal number of a given set and the number of the topologies that can be taken from this set?

I'm a 1st semester sophomore math major right now and I'm taking an Intro Topology course this semester. It ended up being harder than I expected it to be and I'm fairly certain that I will end up with a B/B+ (most likely a B+) in the course due to one bad exam. How bad will this look when I'm...

how do I show a topological space X with an order topology is regular. I've shown it is hausdorff already.

I realize this may be kind of strange, but as it turns out, I've experienced a good introduction to topology (at the level of Munkres) before taking a rigorous class on analysis. With a month-long winter break in my near future, I'm wondering if anyone could suggest a text on real analysis that...

When we say that a metric space (X,d) induces a topology or "every metric space is a topological space in a natural manner" we mean that: A metric space (X,d) can be seen as a topological space (X,τ) where the topology τ consists of all the open sets in the metric space? Which means that all...

I'm asked to find an example of a non-homogeneous topological space. To be honest I'm not really sure where to get started. Intuitively I think I'm looking for a space where one part of it has different topological properties from another location. I just can't think of a well-known space for...

Homework Statement Prove that if X is compact and Y is Hausdorff then a continuous bijection f: X \longrightarrow Y is a homeomorphism. (You may assume that a closed subspace of a compact space is compact, and that an identification space of a compact space is compact). Homework...

Solving equations in differential topology by pen and paper, and in Microsoft word has been tedious and error prone. Can Mathematica help? Mathematica would be required to deal with tensor equations on a pseudo Riemann manifold of four dimensions with complex matrix entries. It should be...

Homework Statement This problem is from Schaum's Outline, chapter 7 #38 i believe. Let f: (0, inf) -> [-1,1] be given as f(x) = sin(1/x), where R is given the usual euclidean metric topology and (0,inf) and [-1,1] are given the relative subspace topology. Show that f is not an open map...

please can you help me to prove this exercise; Prove that: the Euclidean topology R is finer than the cofinite topology on R please answer me as faster as u can I have an exame on monday and I don't know to provethis exercise!

Homework Statement Let U be a topology on the set Z of integers in which every infinite subset is open. Prove that U is the discrete topology, in which every subset is open. Homework Equations Just the definition of discrete topology The Attempt at a Solution I'm not sure where...

Homework Statement Suppose A is an unbounded subspace of a metric space (X,d) (where d is the metric on X). Fix a point b in A let B(b,k)={a in X s.t d(b,a)<k where k>0 is a natural number}. Let A^B(b,k) denote the intersection of the subspace A with the set B(b,k). Then the...

Homework Statement Prove that the set of maps {f in Cs1(M,N)|f is a covering space} is open in the strong topology. Homework Equations The strong topology has as base neighborhoods sets of functions that are disjointly uniformly near f, along with their derivative, on compact subsets...