Problem II. i) Let Tı and T2 be two topologies on the same space X. Suppose...
10. Let T1 and T2 be two topologies on a set X. Then T1 is said to be a finer topology than T2 (and T2 is said to be a coarser topology than T1) if Ti2 T2. Prove that (i) the Euclidean topology R is finer than the finite-closed topology on R; (ii) the identity function f: (X, Ti) -(X, T2) is continuous if and only if TI is a finer topology than T2.
topology Problem 1. (1) Suppose Ti and Tz are two different topologies on a set X. When is the identity map id X X given by id(r) (2) Show that the subspace topology Ty is the smallest topology on YcX for which the inclusion : Y+X is a continuous map. = ra continuous map from (X, Ti) to (X, T2)?
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...
6 6. Let (X, d) be a metric space and T the topology induced on X by d. Let Y be a subset of X and di the metric on Y obtained by restricting d; that is, di(a, b) d(a, b) for all a and b in Y. If T1 is the topology induced orn Y by di and T2 is the subspace topology on Y (induced by T on X), prove that Ti -T2. [This shows that every subspace...
Problem 2.36. Let X be a space and Y CX. Give Y the subspace topology. Describe (in a useful way) the closed sets in Y.
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.
Problem 1. Suppose X is N(12,42) i) what range of values does X take on 68% of the time? ii) what range of values does X take on 95% of the time? Let Y be the sum of three random variables described above. ii) What is the distribution of Y? iv) U is N(10,4) and V is N(8,9). What is the distribution of U-V? Problem 1. Suppose X is N(12,42) i) what range of values does X take on 68%...