Closed Definition and 1000 Threads

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...

There are three kinds of light-cone string diagrams for four closed string interactions. As displayed by fig. 26.10, 26.11 and 26.12 of section 26.5 of Zwiebach's book. For each light-cone string diagram, it is characterized by two parameters, the time difference of the two interaction...

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,...}. Homework Equations N/A The...

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...

Homework Statement Let (X, d) be a metric space. The set Y in X , d(x; y) less than equal to r is called a closed set with radius r centred at point X. Show that a closed ball is a closed set. Homework Equations In a topological space, a set is closed if and only if it coincides...

Homework Statement Show that the intersection of two closed sets is closed. Homework Equations The Attempt at a Solution Let X and Y be closed sets i.e. X and Y are equal to their closure X_ and Y_. Then X\capY is equal to X_\capY_.

Homework Statement I am trying to prove that, if X is compact and Y is closed, X+Y is closed. Both X and Y are sets of real numbers. Homework Equations The Attempt at a Solution I know that a sum of two closed sets isn't necessarily closed. So I presume the key must be the...

By Weierstrass approximation theorem, it seems to be obvious that every continuous function has a power expansion on a closed interval, but I'm not 100% sure about this. Is this genuinely true or there're some counterexamples?

The drawing shows two fully charged capacitors (C1 = 2.00\muF, q1 = 6.00\muC; C2 = 8.00\muF, q=12.0\muC). The switch is closed, and charge flows until equilibrium is reestablished (i.e., until both capacitors have the sam voltage across their plates). Find the resulting voltage across either...

Homework Statement Show that if E is a closed subset of a compact set F, then E is also compact. Homework Equations I'm pretty sure you refer back to the Heine-Borel theorem to do this. "A subset of E of Rk is compact iff it is closed and bounded" The Attempt at a Solution We...

Suppose f:[a,b]--> R and g:[a,b]-->R. Let T={x:f(x)=g(x)} Prove that T is closed. I know that a closed set is one which contains all of its accumulation points. I know that f and g must be uniformly continuous since they have compact domains, that is, closed and bounded domains. Now T is the...

Homework Statement Let E be a nonempty subset of R, and assume that E is both open and closed. Since E is nonempty there is an element a \in E. Denote the set Na(E) = {x > 0|(a-x, a+x) \subsetE} (a) Explain why Na(E) is nonempty. (b) Prove that if x \in Na(E) then [a-x, a+x] \subset...

Homework Statement In my fluid mechanics text, it states that the Net Force due to a uniform pressure acting on a closed surface is zero or: \mathbf{F} = \int_{surface}p(\mathbf{-n})\,dA = 0 \,\,\,\,\,\,\,(1) where n is the unit normal vector and is defined as positive pointing outward from...

Homework Statement Let E and F be 2 non-empty subsets of R^{n}. Define the distance between E and F as follows: d(E,F) = inf_{x\in E , y\in F} | x - y | (a). Give an example of 2 closed sets E and F (which are non-empty subsets of R^n) that satisfy d(E,F) = 0 but the intersection of E...

Homework Statement Consider a closed triangular box resting within a horizontal electric field of magnitude E = 7.80 & 104 N/C as shown in Figure P24.4. Calculate the electric flux through (a) the vertical rectangular surface, (b) the slanted surface, and (c) the entire surface of the box...

The main question: Let S be a subset in Rn which is both open and closed. If S is non-empty, prove that S= Rn. I am allowed to assume Rn is convex. Things I've considered and worked with: The compliment of Rn is an empty set which has no boundaries and therefore neither does Rn. Therefore...

I have been working on this problem for a few hours and am completely stuck. It seems like a simple problem to me but when I attempt it I get nowhere. The problem is: Show that \frac{1}{3}\oint\oint_{S}\vec{r} \cdot d\vec{s} = V where V is the volume enclosed by the closed surface S=...

Homework Statement i) Construct a bounded closed subset of R (reals) with exactly three limit points ii) Construct a bounded closed set E contained in R for which E' (set of limit points of E) is a countable set. Homework Equations Definition of limit point used: Let A be a subset of...

The set \{1,2,3,4,5\ldots\}...is it closed as a subset of \mathbb{R}? I'm thinking "yes," but I'm unsure of myself for some reason. (And yes, this is just the set of positive integers.

if a set A is both open and closed then it is R(set of real numbers) how we may show it in a proper way