# compact

1. ### Proof related with compact sets

Let (X,p) be a compact metric space. Let C be a collection of open subsets of X such that C covers X, i.e., X= Union of Q where Q belongs to C. Prove that there exists a strictly positive real number r, such that for each point a in X, there is a member Q of the collection C such that an open...
2. ### Weak convergence in compact embedding

Let V and W be two Banach spaces. Assume we have V compactly embedded in W, if we have a sequence (x_{n})_{n} in V which converges weakly in V then can we show that it converges strongly in W? If not, what are additional assumptions necessary for this to be true? Thanks a lot for any...
3. ### To show that the image of a continuous function defined on a compact set is nonempty.

Given a continuous function f defined on a compact set A, to show that f(A) is nonempty, the author of my textbook simply says that "every continuous function defined on a compact set reaches a maximum." Can anyone explain it in more detail? Why is that? Or how should I prove its...
4. ### Countably Compact Sets

I'm working on this problem, and I'm not sure how to solve it. Let (X,\Omega) be a topological space and let A \subseteq X be countably compact. Prove that every subset of A that is closed relative to \Omega is also countably compact. I have a similar proof where this problem is stated without...
5. ### compact surfaces question

Let M be a closed, orientable, and bounded surface in R3 a) Prove that the Gauss map on M is surjective b) Let K+(p)= max{0, K(p)} Show that the integral over the surface M, ∫ K+ dA ≥ 4π. Do not use the Gauss Bonnet theorem to prove this
6. ### Subsets of compact spaces

Can anyone provide me with an example of a compact space that has subsets that aren't compact?
7. ### compact subsurfaces of bordered surfaces of infinite genus

Hello, I am a new user in this helpful forum and I start by posting a question that has been troubling me for quite some time now: I want to find a proof for the following proposition: "A bordered surface of infinite genus contains compact subsurfaces of arbitrarily large genus" A few...
8. ### locall compactness and compact operators

I need help understanding an argument. Here is a brief exposition "In reply to: >>Let B be the unit ball in l^2. Then T(B) = B (surjective!). As any open neighbourhood >>U of 0 contains some rB for a scalar r, T(U) contains an sB for some scalar s. As >>locally compact normed spaces are...
9. ### Prove that A is compact

A={0}u{1/n: n is a positive integer}. Prove A is a compact subset of Q w.r.t the standard topology but non-compact w.r.t discrete topology
10. ### Prove that the boundary of S is compact

Let S be a compact subset of a hausdorff space X. Prove that the boundary of S is compact.
11. ### f and its Fourier transform cannot both be compactly supported

I'm trying to prove with Heisenberg inequality that a function f and its Fourier transform cannot both be compactly supported. I've only found on google that it's a well-known fact but i need to understand the proof...
12. ### Compact in R

Hello! I was wondering if someone could clear something up for me.. Is every non-empty, compact subset of R a closed interval in R? It seems to me that this is true. I know that such sets are closed an bounded, but I don't know how to show that they're connected. Any help is much appreciated!
13. ### compact sets

before I write the problem I just want to clarify that this is not a graded assignment but a practice problem designed to help understand the topic. My professor has limited office hours so I am hoping someone can help me reason this out. does the set Q intersection [0, 1] in R have an open...
14. ### help with proving a set to be compact

let K be compact in R^n. show that a + K is also compact, where a + K := { a + x : x in K} what I was thinking is that since K is compact there exists a finite open cover of K. a is a single element and so can also be covered by a finite open cover and the union of the two covers is also...
15. ### Is a compact subset A of a discrete metric space a finite set?

Question: Prove that a compact subset A of a discrete metric space is a finite set. My Attempt: A is a compact subset of a discrete metric space (S,d). Therefore A is both closed and bounded. Since A is bounded there is some Neighborhood N_r(p) such that A is contained in the neighborhood, A...
16. ### Revised: A sequence of non-empty, compact, nested sets converges to its intersection

Proposition 2.4.7 * S is a metric space with metric \rho . {A_n} is a sequence of descending, non-empty, compact sets. Then for \epsilon > 0,\ lim A_n = A = \bigcap A_n in the Hausdorff sense. In that sense, one must show that (1) A \subseteq N_\epsilon (A_n)...
17. ### A sequence of non-empty, compact, nested sets converges to its intersection.

A text* I am reading offers a proof that a sequence of non-empty, compact, nested sets converges to its intersection in the Hausdorff metric. I do not follow the second half of the proof which shows that, in the limit, a member of the sequence is contained in the intersection in the sense that...
18. ### compact support gaussian function

Hi, I am not mathematician. (I am computer science) I need your help to construct a compact support gaussian function It is ok for an interval [a, b]? f(x) = \exp(-0.5\frac{(x-\mu)^2}{\sigma^2})[x/tex] f(x) = \exp(-0.5\frac{(x-\mu)^2}{\sigma^2}) if x \in (a, b) and [tex]f(x) = 0 otherwise...
19. ### Is it possible to integrate dx/dt*e^t in a nice and compact way?

Hello, I need to integrate this based on certain information. I've tried integration by parts, looked in tables, etc., and am pretty sure there's just no neat way to do it. But I want to check with the forum before I give up. \int \frac{dx}{dt} e^t \: dt The information I have: 1) x=x(t)...
20. ### which is compact

which of the following is compact and why? please explain how? 1) {(x,y):|x|<=1, |y|>=2} 2) {(x,y):|x|<=1, |y|^2>=2} 3) {(x,y):x^2+3y^2<=5} 4) {(x,y):x^2<=y^2 +5}