Jet f be continuons one to one m compact metric space X onto a metric space...
* Exercise 10. Let M be a (non-empty) compact metric space and f: M → M a continuous map such that for every ε > 0 there exists x E M such that d(f(x), z) < E. Show that there exists y M such that f()y Hint: consider the map g: MR defined by g(x)=d(f(x),z).] [8 marks]
Let (X, d) be a compact metric space that has at least two elements. Prove that if y ∈ X then fy(x) = d(x, y) is a continuous function fy : X → R.
3. Suppose that (M, ρ) is a compact metric space and f : (M, p)-+ (M,p) is a function such that (Vz, y E M) ρ (z, y) ρ (f (x), f (y)). a. Let x E (M, ρ) and consider the sequence of points {f(n) (X)}n 1 . (Remember: fn) denotes the composition of f with itself, n times, so for each n, f+() rn, k E N) such that ρ (f(m) (x) ,f(n +k) (r)) < ε ....
i) Does Lebesgue lemma hold true in the plane? Justify your answer! ii) Let (X, d1) be a compact metric space and (Y, d2) a metric space. Suppose that f : X → Y is continuous. Use Lebesgue lemma to show that for every > 0 there exists δ > 0 such that if d1(x, y) < δ then d2(f(x), f(y)) < , that is, f is uniformly continuous.
Let (X, d) be a discrete space and let (Y, d′) be any metric space. Prove that any function f : (X, d) → (Y, d′) is continuous. (Namely, any function from a discrete space to any metric space is continuous.)
Let (X, d) be a compact metric space. Prove that if F ⊆ C(X) is equicontinuous then it is uniformly equicontinuous.
Carefully and rigorously prove the following.
Let X be a metric space. Show X is compact if and only if every sequence contains a convergent subse- quence. Hint for (): Argue by contradiction. If there was a sequence with no convergent subsequence, use that sequence to construct an open cover of X, such that every set in the cover contains only a finite number of elements of the sequence. Then use compactness to get a contradiction. Hint for (): Let...
(TOPOLOGY) Prove the following using the defintion:
Exercise 56. Let (M, d) be a metric space and let k be a positive real number. We have shown that the function dk defined by dx(x, y) = kd(x,y) is a metric on M. Let Me denote M with metric d and let M denote M with metric dk. 1. Let f: Md+Mk be defined by f(x) = r. Show that f is continuous. 2. Let g: Mx + Md be defined...
A subset D of a metric space (X, d) is dense if every member of
X is a limit of a sequence of elements from D.
Suppose (X,d) and (Y,ρ) are metric spaces and D is a dense
subset of X.
1. Prove that if f : D -» Y is uniformly continuous then there exists an extension15 of f to a if dn (E D) e X define 7(x) lim f(d,) uniformly continuous function f:X * Y. Hint: 2....
1. Let (X, d) be a metric space, and U, V, W CX subsets of X. (a) (i) Define what it means for U to be open. (ii) Define what it means for V to be closed. (iii) Define what it means for W to be compact. (b) Prove that in a metric space a compact subset is closed.