1- if A1,A2,A3,...,Am are each countable sets, then the union A1 U A2 U A3...U Am is countable
2- if An is a countable set for each n belong to N,then Un=1(to infinity) An is countable

1. Show that if A and B are countableand disjoint, then A U B is countable.
2. Show that any set, A, of cardinality c contains a subset, B, that is denumerable.
3. Show that the irrational numbers have a cardinality c.
4. Show that if A is equivalent to B and C is equivalent to D, then A x C is equivalent to B x D.

Please help with the following problem.
Use the following steps to prove that every non-empty open subset of R is a union of at most countably many disjoint open intervals.
Suppose that G is a non-empty open subset of R.
1. For each a <- G let Ia be the union of all those open intervals I which contain a and are containe

a) If {I } is a finite or countable collection of disjoint open intervals with
I (a, b), Prove that (m: Measure).
b) If U (empty set) is an open set of R, Prove That (U)> 0.
c) Let P denote the cantor set in [0, 1]. Prove that m(P) = 0.
d) Suppose that U, V are open subsets of R, a,b R with U [a,b] = V [a,

(See attached file for full problem description with all symbols)
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2.14 (I) Prove that an infinite set X is countable if and only if there is a sequence
of all the elements of X which has no repetitions.
(II) Prove that every subset S of a countable set X is itself countable.
(III) Prove that if

2. Let X be given the co-finite topology. Suppose that X is a infinite set. Is X Hausdorff? Should compact subsets of X be closed? What are compact subsets of X?
3. Let (X,T) be a co-countable topological space. Show that X is connected if it is uncountable. In fact, show that every uncountable subspace of X is connected.

For any set B, let P(B) denote the power set of B (the collection of all subsets of B):
P(B) = {E: E is a subset of B}
Let A be a countably infinite set (an infinite set which is countable), and do the following:
(a) Prove that there is a one-to-one correspondence between P(A) and the set S of all countably infinite seq

Let A be a sigma algebra of subsets of R (Real numbers) and suppose I is a closed interval which is in A. Let A(I) denote the collection of all subsets of I which are in A. prove that A(I) is a sigma algebra of subsets of I.

Let {x_alpha}_[(alpha)(E)(gamma)] be an indexed collection of non-negative real numbers. The sum of this collection is defined to be the supremum of the set of all sums over finite subsets of gamma. This is
The sum of_[(alpha)(E)(gamma)]x_(alpha) = sup{(the sum of)_(alpha)(E)(S)x_(alpha): S is finite subset of gamma}
Prove t