Let A be a collection of sets, then We defined A’ = Union of all elements...

Let A be a collection of sets, then We defined A’ = Union of all elements of A.

Definition: If A = NOT the union of C and D , where C and D are non empty sub-collection,

such that C’ intersect D’ = empty

Then A is coherent.

Prove: if A is coherent then A’ is connected (i.e A’ is not the union of two separated sets in standard topology)

Expert Solution

Consider the contrapositive statement " if A' is disconnected then A is not coherent." It is enough to prove this contrapositive statement for our proof.

For let A' be disconnected, then there exist nonempty subsets B' and C' of A' such that A' = C' U B' and C' B' = . Correspondingly there exists nonempty sub collections C and B of A such that C' and B' are the union of all elements of C and B respectively. Clearly C U B A. Now let a A then a A' either a C' or a B' but not in both.

either a C or a B

a C U B

A C U B

A = C U B.

Hence there exists nonempty sub collections C and B of A such that A = C U B and C' B' = . Therefore A is not coherent.

Therefore if A' is disconnected then A is not coherent, which implies if A is coherent then A' is connected.

Colby Messinger answered 2 years ago

(a) Give an example of a collection of closed sets whose union is not closed. (b)...

(a) Give an example of a collection of closed sets whose union is not closed. (b) Give an example of a collection of open sets whose infinite intersection is not open. Thank you!

Let Ω be any set and let F be the collection of all subsets of Ω...

Let Ω be any set and let F be the collection of all subsets of Ω that are either countable or have a countable complement. (Recall that a set is countable if it is either finite or can be placed in one-to-one correspondence with the natural numbers N = {1, 2, . . .}.) (a) Show that F is a σ-algebra. (b) Show that the set function given by μ(E)= 0 if E is countable ; μ(E) = ∞ otherwise...

The intersection (∩) of two sets (s1, s2) is the set of all elements that are...

The intersection (∩) of two sets (s1, s2) is the set of all elements that are in s1 and are also in s2. Write a function (intersect) that takes two lists as input (you can assume they have no duplicate elements), and returns the intersection of those two sets (as a list) without using the in operator or any built-in functions, except for range() and len(). Write some code to test your function, as well. Note: Do not use the...

we have defined open sets in R: for any a ∈ R, there is sigma >...

we have defined open sets in R: for any a ∈ R, there is sigma > 0 such that (a − sigma, a + sigma) ⊆ A. (i) Let A and B be two open sets in R. Show that A ∩ B is open. (ii) Let {Aα}α∈I be a family of open sets in R. Show that ∪(α∈I)Aα is open. Hint: Follow the definition of open sets. Please be specific and rigorous! Thanks!

C++ Given two finite sets, list all elements in the Cartesian product of these two sets.

Let A and B be sets. Then we denote the set of functions with domain A...

Let A and B be sets. Then we denote the set of functions with domain A and codomain B as B^A. In other words, an element f∈B^A is a function f:A→B. Prove: Let ?,?∈?^? (that is, ? and ? are real-valued functions with domain ?) and define a relation ≡ on ?^? by ?≡?⟺?(0)=?(0). (That is, ? and ? are equivalent if and only if they share the same value at ?=0) Then ≡ is an equivalence relation on ?^?. Prove:...

Prove that the union of a finite collection of compact subsets is compact

Let ? be the sample space of an experiment and let ℱ be a collection of...

Let ? be the sample space of an experiment and let ℱ be a collection of subsets of ?. a) What properties must ℱ have if we are to construct a probability measure on (?,ℱ)? b) Assume ℱ has the properties in part (a). Let ? be a function that maps the elements of ℱ onto ℝ such that i) ?(?) ≥ 0 , ∀ ? ∈ ℱ ii) ?(?) = 1 and iii) If ?1 , ?2 … are...

The statement: "int ar[7] = {0};" sets all of the array elements of ar to 0...

The statement: "int ar[7] = {0};" sets all of the array elements of ar to 0 (zero) True False ------------------------------------------- #define SIZE 3 - declares a constant value named SIZE, equal to 3, that is available throughout your program and is immutable. True False ---------------------------------------- Select all of the following that apply to an array: 1. Contiguous storage is storage without any gaps. 2. Arrays make programming complex problems more difficult. 3. An arrays elements are stored contiguously in memory....

Prove that the union of infinitely many countable sets is countable.

Question