Question

In: Advanced Math

Let X = NN endowed with the product topology. For x ∈ X denote x by...

Let X = NN endowed with the product topology. For x ∈ X denote x by (x1, x2, x3, . . .).

(a) Decide if the function given by d : X × X → R is a metric on X where, d(x, x) = 0 and if x is not equal to y then d(x, y) = 1/n where n is the least value for which xn is not equal to yn. Prove your answer.

(b) Show that no compact set in X can contain a non-empty open set.

Solutions

Expert Solution


Related Solutions

Suppose two fair dice are rolled. Let X denote the product of the values on the...
Suppose two fair dice are rolled. Let X denote the product of the values on the dice and Y denote minimum of the two dice. Find E[X] and E[Y] Find Var X and Var Y Let Z=XY. Find E[Z]. Find Cov(X,Y) and Corr(X,Y) Find E[X|Y=1] and E[Y|X=1]
Exercise 31: (General definition of a topology) Let X be a set and O ⊂ P(X),...
Exercise 31: (General definition of a topology) Let X be a set and O ⊂ P(X), where P(X) := {U ⊂ X}. O is a topology on X iff O satisfies (i) X∈O and ∅∈O; (ii) ?i∈I Ui ∈ O where Ui ∈ O for all i ∈ I and I is an arbitrary index set; (iii) ?i∈J Ui ∈ O where Ui ∈ O for all i ∈ J and J is a finite index set. In a general...
TOPOLOGY Let f : X → Y be a function. Prove that f is one-to-one and...
TOPOLOGY Let f : X → Y be a function. Prove that f is one-to-one and onto if and only if f[A^c] = (f[A])^c for every subset A of X. (prove both directions)
Let R be a ring (not necessarily commutative), and let X denote the set of two-sided...
Let R be a ring (not necessarily commutative), and let X denote the set of two-sided ideals of R. (i) Show that X is a poset with respect to to set-theoretic inclusion, ⊂. (ii) Show that with respect to the operations I ∩ J and I + J (candidates for meet and join; remember that I+J consists of the set of sums, {i + j} where i ∈ I and j ∈ J) respectively, X is a lattice. (iii) Give...
Will Smith consumes chocolate and milk. Let y denote chocolate and x denote milk. Chocolate has...
Will Smith consumes chocolate and milk. Let y denote chocolate and x denote milk. Chocolate has an unusual market where there is only one supplier, and the more chocolate you buy from the supplier, the higher the price she charges per unit. In fact, y units of chocolate will cost Will y2 dollars. Milk is sold in the usual way at a price of 2 dollars per unit. Will’s income is 20 dollars and his utility function is U =...
A fair 4-sided die is rolled, let X denote the outcome. After that, if X =...
A fair 4-sided die is rolled, let X denote the outcome. After that, if X = x, then x fair coins are tossed, let Y denote the number of Tails observed. a) Find P( X >= 3 | Y = 0 ). b) Find E( X | Y = 2 ). “Hint”: Construct the joint probability distribution for ( X, Y ) first. Write it in the form of a rectangular array with x = 1, 2, 3, 4 and...
Let X be a non-degenerate ordered set with the order topology. A non-degenerate set is a...
Let X be a non-degenerate ordered set with the order topology. A non-degenerate set is a set with more than one element. Show the following: (1) every open interval is open, (2) every closed interval is closed, (3) every open ray is open, and (4) every closed ray is closed. Please note: Its a topology question.
Prove that the discrete topology on X is the same as the metric topology induced by...
Prove that the discrete topology on X is the same as the metric topology induced by the discrete metric. Where metric topology is defined as: If (X,d) is a metric space, then consider the collection T of all open subsets of X. Then (X,T) is topological space. This topology is called the metric topology on X induced by d.
A manufacturer produces lots of a canned food product. Let p denote the proportion of the...
A manufacturer produces lots of a canned food product. Let p denote the proportion of the lots that do not meet the product quality specifications. An n = 29, c = 0 acceptance sampling plan will be used. (a) Compute points on the operating characteristic curve when p = 0.01, 0.03, 0.10, and 0.20. (Round your answers to four decimal places.) c p = 0.01 p = 0.03 p = 0.10 p = 0.20 0 (c) What is the probability...
A manufacturer produces lots of a canned food product. Let p denote the proportion of the...
A manufacturer produces lots of a canned food product. Let p denote the proportion of the lots that do not meet the product quality specifications. An n = 25, c = 0 acceptance sampling plan will be used. (a) Compute points on the operating characteristic curve when p = 0.01, 0.03, 0.10, and 0.20. (Round your answers to four decimal places.) c p = 0.01 p = 0.03 p = 0.10 p = 0.20 0 What is the probability that...
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT