Topology Questions?

Question

1. (a) show how to write N(natural numbers) as the union of infinitely many infinite pairwise disjoint subsets.
(b) show how to write N(natural numbers) as the union of uncountably many sets with the property that, given any two of them, one is a subset of the other.

2. Let D be a countable set of points in the plane R^2,
Prove there exist sets A and B such that D = A U B,
where the set A has finite intersection with every horizontal line in the plane and B has finite intersection with every vertical line in the plane.

Answer 1

1a. For each prime p, let S_p be the set of all positive integers having p as their smallest prime factor - except for S_2 which will be defined to include 1 as well (and 0 if your definition of natural numbers includes it). Then clearly all such sets are infinite, disjoint and include all positive integers.

1b. For each real number a > 0, define R_a to be the set of all rational numbers in the interval (-a,a). Clearly the number of such sets is uncountable and for any two positive real numbers a,b either R_a is contained in R_b or vice versa. Since there is a bijection between the rational numbers and the natural numbers, use it to define a corresponding collection of sets of natural numbers with exactly the same properties.


2. Since D is countable, the set of all distinct x coordinate values is also countable, as is the set of distinct y coordinate values. That means each point p in D can be written p = (x_i,y_j), where i,j = 1,2,3....

Assign each p to A if and only if i <= j, otherwise assign it to B. Then every horizontal line containing points of D has a unique x coordinate x_k and can have at most k distinct y values (i.e. k members of D). Likewise each vertical line can contain only finitely many members of B.

Answer 2

1. is kinda drawn out. Here's a sketch: Partition N into n subsete with the property that sup(x_a) = inf(x_a+1) and show that the limit (as n -> ∞) = N

2. Let A = { p,y | a_0 ≤ p ≤a_1, y ε Y} (Y is set of all values on the vertical axis)
and B = {x,q | b_0 ≤ q ≤b_1 | x ε X} (X is set of all values on the horizontal axis)

Clearly, for all p and q, p ε X and q ε Y so that D = {p,q} is the desired set.


Doug