Question

(a) If G is an open set and K is a compact set with K ⊆...

(a) If G is an open set and K is a compact set with K ⊆ G, show that there exists a δ > 0 such that {x|dist(x, K) < δ} ⊆ G.

(b) Find an example of an open subset G in a metric space X and a closed, non compact subset F of G such that there is no δ > 0 with {x|dist(x, F) < δ} ⊆ G

Math   Advanced-Math   
0 0
Add a commentTranscribed image text
Next >

Homework Answers

Answer #1

0 0
Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Log In

Sign Up

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
Suppose K is a nonempty compact subset of a metric space X and x∈X. Show, there...
Suppose K is a nonempty compact subset of a metric space X and x∈X. Show, there is a nearest point p∈K to x; that is, there is a point p∈K such that, for all other q∈K, d(p,x)≤d(q,x). [Suggestion: As a start, let S={d(x,y):y∈K} and show there is a sequence (qn) from K such that the numerical sequence (d(x,qn)) converges to inf(S).] Let X=R^2 and T={(x,y):x^2+y^2=1}. Show, there is a point z∈X and distinct points a,b∈T that are nearest points to...
Let (X,d) be a metric space which contains an infinite countable set Ewith the property x,y...
Let (X,d) be a metric space which contains an infinite countable set Ewith the property x,y ∈ E ⇒ d(x,y) = 1. (a) Show E is a closed and bounded subset of X. (b) Show E is not compact. (c) Explain why E cannot be a subset of Rn for any n.
(2) If K is a subset of (X,d), show that K is compact if and only...
(2) If K is a subset of (X,d), show that K is compact if and only if every cover of K by relatively open subsets of K has a finite subcover.
Show that if Y is G-delta set in X , and if X is a compact...
Show that if Y is G-delta set in X , and if X is a compact Hausdorff space, then Y is a Baire space in subspace topology.
Let (X, d) be a compact metric space and let A ⊆ X. Suppose that A...
Let (X, d) be a compact metric space and let A ⊆ X. Suppose that A is not compact. Prove that there exists a continuous function f : A → R, from (A, d) to (R, d|·|), which is not uniformly continuous.
These problems concern the discrete metric. You can assume that the underlying space is R. (a)...
These problems concern the discrete metric. You can assume that the underlying space is R. (a) What does a convergent sequence look like in the discrete metric? (b) Show that the discrete metric yields a counterexample to the claim that every bounded sequence has a convergent subsequence. (c) What does an open set look like in the discrete metric? A closed set? (d) What does a (sequentially or topologically) compact set look like in the discrete metric? (e) Show that...
Supply proofs for the following miscellaneous propositions from the course in a metric space context: (e)...
Supply proofs for the following miscellaneous propositions from the course in a metric space context: (e) A set is open if and only if its complement is closed. (f) A compact set (you may use either definition) is closed and bounded.
Let (X, d) be a metric space, and let U denote the set of all uniformly...
Let (X, d) be a metric space, and let U denote the set of all uniformly continuous functions from X into R. (a) If f,g ∈ U and we define (f + g) : X → R by (f + g)(x) = f(x) + g(x) for all x in X, show that f+g∈U. In words,U is a vector space over R. (b)If f,g∈U and we define (fg) : X → R by (fg)(x) = f(x)g(x) for all x in X,...
Let (X, d) be a compact metric space and F: X--> X be a function such...
Let (X, d) be a compact metric space and F: X--> X be a function such that d(F(x), F(y)) < d(x, y). Let G: X --> R be a function such that G(x) = d(F(x), x). Prove G is continuous (assume that it is proved that F is continuous).
Problem 6. For a closed convex nonempty subset K of a Hilbert space H and x...
Problem 6. For a closed convex nonempty subset K of a Hilbert space H and x ∈ H, denote by P x ∈ K a unique closest point to x among points in K, i.e. P x ∈ K such that ||P x − x|| ≤ ||y − x||, for all y ∈ K. First show that such point P x exists and unique. Next prove that all x, y ∈ H ||P x − P y|| ≤ ||x −...
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT