# compactness

1. ### Easy compactness proof, What is wrong with my picture/logic?

Theorem. If {Kα} is a collection of compact subsets of a metric space X such that the intersection of every finite subcollection of {Kα} is nonempty, then ∩Kα is nonempty. Proof. Fix a member K1 of {Kα} and put Gα=Kcα [this denotes the complement of Kα]. Assume that no point of K1 belongs...
2. ### Questions in mathematical logic - compactness theorem

① L1 = <S> is a lexicon (alphabet). T1 is a set of sentences (theory) in L1 that have true value in the structure (model) M = [{1,2,3}, <]. We define L2 = L1 ∪ [S, c1, c2, c3, e]. We define T2 = T1 ∪ {c1<e, c2<e, c3<e}. Is T2 consistent? Why? ② L1 = <S, c1, c2, c3> is a lexicon. T1 is a set of...
3. ### Compactness of the operator

Hello there, I'd like to ask for help with this exercise: Let X=L^2(\mathbb{R}) and \varphi : \mathbb{R}\rightarrow\mathbb{R} continuous function, for which there holds \lim_{x\to\pm\infty}f(x)=0 and \exists x\in \mathbb{R}\,:\, \varphi (x)\neq 0 . In addition let be A:X\rightarrow X linear...
4. ### Compactness and Product Topologies

So, I have a problem that I'm working on and I can't seem to figure it out. We just started product topologies, so many properties are still new or unknown to me. Here's the problem: Let (X,\Omega) and (Y,\Theta) be topological spaces. If A \subseteq Y is compact relative to \Theta and x \in X...
5. ### Applying compactness theorem in First Order Logic

Hey! Sorry for posting again so quickly, but I also have an issue for a second concept. This one is regarding applying the compactness theorem in first order logic, the framework is a Hilbert system: L is a FOL (First order language) with R, where R is a single binary predicate symbol...
6. ### 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...
7. ### Compactness of topological space

Hey. My assignment says the following: Let X be a topolical space with \tau as its topology. Let \infty be a point not in X. Let X^* = X \cup \{ \infty \} . Let \tau^* = \tau \cup \{ U \in X | X^* \setminus U is a closed, compact subset of X \}. (1) Prove that \tau^* is a topology on...
8. ### Compactness and sequentiall compactness

is there a difference between compactness and sequential compactness in a metric space or are these two terms simply interchangable? From what i can understand sequentially compact means that every subsequence has a convergent subsequence within the space and compact is the topological...
9. ### Corollary of compactness theorem

Show that if a set of sentences S entails a sentence F, then some finite subset of S implies F. Here's what I have so far. Suppose S entails F, and for reductio, that no finite subset of S implies F. So, for every finite subset S*, there is an interpretation that makes every sentence in S*...
10. ### Compactness over discrete metric space

Any idea that help in solving this problem is highly appreciated :)
11. ### compactness of sets of homeomorphisms of a group.

Hello; Let M be a manifold and let U be a nonempty euclidean open subset of M. Let B be a nonempty open ball whose compact closure cl(B) is contained in U. Let H(cl(B)) be the group of homeomorphisms of cl(B). Then prove that the set of all functions in H(cl(B)) that fix the boundary ∂B of B...
12. ### Limit point compactness

Hi, I have a question about an exercise in Munkres, \S 28, 3. At first I thought I had proven some of these things, but I've seen some counterexamples online and I'm not familiar with some of the spaces used. Let X be limit point compact. (b) If A is a closed subset of X, does it...
13. ### show compactness

Let \mathbf{X}=L^2(\mathbf{N})=\{\ x=\{\ x_n \}\ | \sum_{n=1} ^\infty = x_n ^2 < \infty \}, \| x \| = ( \sum_{n=1} ^\infty x _n ^2 )^{1/2} and \mathbf{K} \subset \math{X} be closed and bounded with the property \forall \epsilon > 0 there exists N such that \forall n >N, \sum_{n=N} ^\infty x_n...
14. ### Lebesgue's Number Lemma

Hello there! I have a fundamental (I suppose) question on Analysis. What is the Lebesgue's Number Lemma? What is all about? I read it but I can't figure out what it practically means. Could anybody explain to me?
15. ### compactness and connectedness in R^2

i am just confused how to solve it. Let A={(x, x Sin(1/x)) ∈ R²:x ∈ (0,1] } and B=A U (0,0). Then 1. is A compact and connected? 2. is B compact and connected? Thanks in advance. Regards(Happy)(Happy)
16. ### is the set A={ (x,y)∈R² : x²+ y² =1} connected?

is the set A={ (x,y)∈R² : x²+ y² =1} connected? how do we prove it? is there any general proof? or some quick method by which we can tell any set is connected or not
17. ### Compactness

Consider the next set. \Lambda = \left\{ {f \in C\left( {\left[ {0,1} \right],\mathbb{R}} \right);\left| {f\left( x \right) - f\left( y \right)} \right| \leqslant \pi \left| {x - y} \right| \wedge \left\| f \right\|_\infty \leqslant 1} \right\} I have been ask to prove that \Lambda in...
18. ### topology- compactness, interior=empty

I need help with this exercise..(Thinking) Let \{ X_\alpha\}_{\alpha \in I} be a collection of topological spaces. X=\prod_{\alpha \in I}X_\alpha 1)prove: If an infinite number of X_\alpha are non-compact, then any compact subset in \prod_{\alpha \in I}X_\alpha has empty interior. 2)is there a...
19. ### Local Compactness

Dear Colleagues, A metric space X is said to be locally compact if every point of X has a compact neighborhood. Show that a compact metric space X is locally compact. Regards, Raed.
20. ### Compactness Theorem and partial orderings

Hello folks, I've been asked 'If P is a partial ordering, how do I use the compactness theorem to show that P is the union of k chains iff each finite subset on P is the union of k chains?' But, I have absolutely no idea what this question is even driving at. Set theory and logic is easily...