Closed Definition and 1000 Threads

Homework Statement A = \left\{(x,y): 0\leq xy \leq 1\right\}, A \in R^{2} I'm trying to determine if this set is bounded and/or closed. Homework Equations if X = (x,y) euclidean metric: ||X|| = \sqrt{x^{2}+y^{2}} The Attempt at a Solution I know a bounded set =>...

Homework Statement on the complex line, with the usual metric, I need to determine if this is a closed set. A = \left\{\left|\frac{1}{z^{2}+1} \right|: |z| = 1 ; z\neq \pm i\right \} Homework Equations The Attempt at a Solution A closed set implies that the set of all limit points belongs...

Homework Statement Let (X,\tau_X) and (Y,\tau_Y) be topological spaces, and let f : X \to Y be continuous. Let Y be Hausdorff, and prove that the graph of f i.e. \graph(f) := \{ (x,f(x)) | x \in X \} is a closed subset of X \times Y. Homework Equations The Attempt at a Solution...

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 l∞ be the space of bounded sequences of real numbers, endowed with the norm ∥x∥∞ = supn∈N |xn | , where x = (xn )n∈N . Prove that the closed unit ball of l∞ , B(0, 1) = {x ∈ l∞ ; ∥x∥∞ ≤ 1} , is not compact. Homework Equations The Attempt at a Solution I'm...

Apostol page 386, problem 5 Homework Statement Given f,g continuously differentiable on open connected S in the plane, show \oint_C{f\nabla g\cdot d\alpha}=-\oint_C{g\nabla f\cdot d\alpha} for any piecewise Jordan curve C. Homework Equations 1. Green's Theorem 2. \frac{\partial...

If (X,\tau) is either a T_1 space or Hausdorff space then for any x \in X the singleton set \{ x \} is closed. Why is this the case? I can't see the reason from the definitions of the spaces. Definition: Let (X,\tau) be a topological space and let x,y \in X be any two distinct points, if...

Homework Statement A projecticle launcher is shown in the attachment. A large current moves in a closed loop composed of fixed rails, a power supply, and a very light, almost frictionless bar touching the rails. A magnetic field is perpendicular to the plane of the circuit. If the bar has a...

Homework Statement An organ pipe has two successive harmonics with frequencies 1760 Hz and 2160 Hz. Is this an open or stopped pipe? Which two harmonics are these? What is the length of the pipe? Homework Equations L=v/2f The Attempt at a Solution I'm really just having trouble...

Homework Statement a) Let V be a normed vector space. Then show that (by the triangle inequality) the function f(x)=||x|| is a Lipschitz function from V into [0,∞). In particular, f is uniformly continuous on V. b) Show that a closed subset F of contains an element of minimal norm, that...

I understand that magnetic flux through a closed surface is zero, but what is the exact definition of a closed surface? The textbook I'm using is rather vague with this definition and I want to make sure I have the definition nailed down for the exam in case my professor tries anything tricky.

So confused about standing waves in a closed pipe, which is open at one end and closed at the other. The closed end has a node while the open end has an antinode. To figure the wavelength, i use the formula: Lambda = 4L/n where n is the number of harmonic and can only be odd integers...

I'm reading Riley's "Mathematical Methods for Physics and Engineering" and I came across this expression about vector spaces: "A set of objects (vectors) a, b, c, ... is said to form a linear vector space V if the set is closed under commutative and associative addition (...)" What I don't...

We know that a linear operator T:X\rightarrowY between two Banach Spaces X and Y is an open mapping if T is surjective. Here open mapping means that T sends open subsets of X to open subsets of Y. Prove that if T is an open mapping between two Banach Spaces then it is not necessarily a closed...

[b]1. the lowest note on an organ is 16.4 Hz. What is the shortest open organ pipe that will resonate at this frequnecy? What would be the pitch if the same organ pipe were closed? [b]3. is the answer 32.8 meters and 65.6 meters?

I think cos(x) is closed function in R. But I heard that cos(x) is not closed function in R. What do I choose closed set A in R, cos(A) is not closed in R? Help...

how can i calculate density of a liquid pumped in a closed circuit and i don't know the nature of it ?

Homework Statement Consider the power series. sigma (n=1 to infinity) x^n / [n(n+1)] if f(x) = sigma x^n / [n(n+1)], then compute a closed-form expression for f(x). It says: "Hint: let g(x) = x * f(x) and compute g''(x). Integrate this twice to get back to g(x) and hence derive...

how i prove that space of defrential function not closed?

If a set A and a set B are equivalent, and it is known that A is complete, can it then be said that B is also complete? What if it is known that A is closed, can it then be said that B is also closed?

This isn't actually a homework/coursework question, but rather a need to clarify a discepancy between my lecturer's notes and a textbook. My lecturer's notes state that for an "organ" pipe, closed at one end, the 1st harmonic frequency will be 4L. For the 2nd harmonic the frequency will be...

Hi all, for a closed container with water under pressure in it, let's say GAUGE PRESSURE of the container is 101 kPa, what will the boiling point of water in that vessel be? Will it be 100 deg C, or will it be 120 deg C because the ABSOLUTE PRESSURE of the water in that vessel is 202kPa...

I have a problem with this question. The question is "Find closed formula for the sequence 0,1,3,0,1,3,0,1,3..." It can be written as (0,1,0,0,1,0,0,1,0...)+3*(0,0,1,0,0,1,0,0,1...) I know the sequence of 0,0,1,0,0,1..., but I do not know how to get the sequence of 0,1,0,0,1,0... And is...

I'm sorry if this should be in the Analysis forum; I figured it pertained to topology though. Let Y be a subspace of a metric space (X,d) and let A be a subset of Y. The proposition includes conditions for A to be open or closed in Y. In class the teacher first proved when A is open and then...

Definition: Let F be a subset of a metric space X. F is called closed if whenever is a sequence in F which converges to a E X, then a E F. (i.e. F contains all limits of sequences in F) The closure of F is the set of all limits of sequences in F. Claim 1: F is contained in the clousre of F...

In my cosmology lectures they say that a negative curvature gives an infinite space but I was thinking what about the inside of a torus. Isn't that a closed space too? Cant any value of k apart from 0 result in a closed space??

"Closed" set in a metric space Homework Statement 1) Let (X,d) be a metric space. Prove that a "closed" ball {x E X: d(x,a) ≤ r} is a closed set. [SOLVED] 2) Suppose that (xn) is a sequence in a metric space X such that lim xn = a exists. Prove that {xn: n E N} U {a} is a closed subset of...

Consider a 4 dimensional spacetime which is everywhere flat and is closed in along all three spatial dimensions. Since spacetime is everywhere flat we can use global inertial frames. We will not consider any gravitational interactions in this problem. Now a valid vacuum solution to maxwell's...

Homework Statement Is [0, infinity) a closed set? Homework Equations N/A The Attempt at a Solution It's easy to say that its not. But the solution in my textbook suggests otherwise. Why is this so? Thanks! M

Im working through derivations of string equations of motion from the Nambu-Goto Action and I'm stuck on something that I think must be trivial, just a math step that I can't really see how to work through. At this point I've derived the equation of motion for the closed string from the wave...

Homework Statement Give an example of a decreasing sequence of closed balls in a complete metric space with empty intersection. Hint 1: use a metric on N topologically equivalent to the discrete metric so that {n≥k} are closed balls. In={n,n+1,n+2,...}.[/color] Homework Equations N/A...

I`ve thought about a special sort of twin paradox. I know the usual explanation of the twin paradox but give me please the answer to this special case: Imagine: A static universe (non-expanding) with a closed geometry and a circumference of one lightyear. The twins start their journey in...

Homework Statement I'm having a problem with the circuit in the attached diagram. I am looking for the charge on the plates of the capacitor a long time after the switch is closed. Homework Equations The Attempt at a Solution I found the current leaving the battery is 0.962 A a...

Let f: D->R be continuous. If D is not open, then f(D) is not open. Why can they not replace 'not open' with closed? Thank you M

Hi, everyone: I would appreciate any help with the following: I am trying to refresh my knot theory--it's been a while. I am trying to answer the following: 1) Every simple polygonal knot P in R^2 is trivial.: I have tried to actually construct a...

Show the progression (6k +1) (k is an integer) is closed under multiplication: Firstly I should check that I remember what this means... If it is closed when you multiply any 2 elements together you get an element that is in the set? So for this I thought just show (6k+1)(6n+1), where k...

Hello, my name is Edward Solomon, after much experimentation and calculation I have failed to make a system that can reflect light in a closed system. Now I am not naive enough to believe I can make a true closed system. There is an absorption and conversion to heat each time light strikes a...

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...

A basic question: What does "closed form" mean? "The point here is that \sigma algebras are difficult but \pi systems are easy: one can often write down in closed form the general element of a \pi system while the general element event of \mathbf B \mathbf is impossibly complicated" - From the...

Homework Statement This is the Theorem as stated in the book: Let S be a subset of a metric space E. Then S is closed if and only if, whenever p1, p2, p3,... is a sequence of points of S that is convergent in E, we have: lim(n->inf)pn is in S. Homework Equations From "introduction to...

Homework Statement Hi all I wish to show differentiability of g(x)=x on the interval [-pi, pi]. This is what I have done: g'(a) = \mathop {\lim }\limits_{h \to 0} \frac{{g\left( {a + h} \right) - g\left( {a} \right)}}{h} \\ = \mathop {\lim }\limits_{h \to 0} \frac{h}{h} \\ = 1...

Prove that countable intersections of closed subset of R^d are closed

Homework Statement Hi guys, this problem gave me some trouble before, but I'd like to know if I have it worked out now... "If S = S\cupBdyS, then S is closed (S_{compliment} is open) Homework Equations S is equal to it's closure. The Attempt at a Solution 1. Pick a point p in...

Hi, I'm studying for the final exam in my first course in topology. I'm currently recalling as many theorems as I can and trying to prove them without referring to a text or notes. I think I have a proof that the closed interval [0,1] is connected, but it's different than what I have in my...

Homework Statement A closed system comprising a cylinder and frictionless piston contains 1kg of a perfect gas of which molecular mass is 26. The piston is loaded so that the pressure is constant at 200kPa. Heat is supplied causing the gas to expand from 0.5m^3 to 1m^3. Calculate heat...

Homework Statement Prove that a closed rectangle A \subset \mathbb{R}^n is a closed set. Homework Equations N/A The Attempt at a Solution Let A = [a_1,b_1] \times \dots \times [a_n,b_n] \subset \mathbb{R}^n, then A is closed if and only if its complement, \mathbb{R}^n - A, is...

Homework Statement The container shown in the figure is filled with oil. It is open to the atmosphere on the left. What is the gauge pressure at point A? Point A is 50cm high from the ground and 50 cm from the top. Homework Equations po+(density)gd The Attempt at a Solution I...

Okay, so I'm trying to finish of a problem on integral closure and I am rather unsure if the following fact is true: If L embeds into an algebraically closed field K and F is an algebraic extension of L, then it is possible to extend the embedding of L to F into K. Now the case where F...

Homework Statement Using Gauss's theorem prove that \int_{s}\vec{n}ds=0 if s is a closed surface. Homework Equations Gauss's theorem: \int_{V}\nabla.\vec{A}dv= \oint_{s}\vec{A}.\vec{n}da The Attempt at a Solution In this problem \vec{A} is constant so \nabla.\vec{A}=0 so...

Homework Statement Let X be set donoted by the discrete metrics d(x; y) =(0 if x = y; 1 if x not equal y: (a) Show that any sub set Y of X is open in X (b) Show that any sub set Y of X is closed in y Homework Equations In a topological space, a set is closed if and only if it...