Question 3. Consider the usual topology U on the set of real numbers. Describe the relative topology UN on the set N of positive integers.

Let R be the relation on the set of all mathematicians that contains the ordered pair (a, b) if and only if a and b have written a published mathematical paper together. a) Describe the relation R[2] in the context of the problem.b) Describe the relation R* in the context of the problem.c) The Erdos number of a mathematician is 1 if this mathematician wrote a paper with the prolific Hungarian mathematician Paul Erdos. The Erdos number is 2 if a mathematician didn’t write a joint paper with Erdos but wrote a joint paper with someone who wrote a joint paper with Erdos, and so on (except that the Erdos number of Erdos himself is 0). Give a definition of the Erdos number is terms of paths in R. List a name of a mathematician with number: 1, 2, and 3

please prove theorem 1.13 from topology through inquiry and provide any theorems or definitions you use

Define the real numbers R to be the set of all cuts in Q.

Show that the translate of an Fδ set is also Fδ.