MTG 4302/5316 Dec. 2012
SOLUTIONS
FINAL
1
1. (a) Is the set C = cfw_(x, y ) R2 | y = x closed?
Yes, since C is the inverse image of a point C = f 1 (cfw_1) for a
continuous function f : R2 R dened by the formula f (x, y ) = xy .
Note that the set cfw_1 i

MTG 4302/5316 Due: Dec. 5, 12:50, 2012
NAME:
FINAL
1
1. (a) Is the set C = cfw_(x, y ) R2 | y = x closed?
(b) connected ?
(c) compact ?
(d) homeomorphic to R2 ?
(e) discrete in the subspace topology inherited from Rl Rl ?
Justify your answer.
2
2. Find t

Topology, MTG 4302/5316 Fall-2012
NAME:
Pop-Quiz solutions
1. Prove that the set of transcendental numbers is uncountable?
Pf. The set of algebraic numbers A is countable since each polynomials has nitely many roots and the number of polynomials with rati

MTG 4302/5316 MidTerm 1
NAME:
1. Prove that the set of all irrational numbers is uncountable.
Solution: If we assume that I is countable, then R = Q I must be
countable as the union of two countable sets. But we know that R is
uncountable. Therefore I is

MTG 4302/5316 Fall-2012 MT2
SOLUTIONS
1
1. Is the subspace C = cfw_(x, y ) R2 | y = x , x > 0 R2
(a) closed in R2 ?
Yes, since C is the intersection of two closed sets C = f 1 (1)
[0, ) R R where the rst is closed as the preimage of a closed
set cfw_1 fo

MTG 4302/5316 Fall-2016 Midterm, SOLUTIONS:
1. Prove that the set of irrational numbers is uncountable.
Assume that it is countable. Then the set of reals will be countable as the union of two countable sets rational numbers and irrational
numbers. Contra

Topology, MTG 4302/5316 Due: Dec. 5, 12:50, 2016
NAME:
FINAL
1. (a) Is the set C = cfw_(x, y) R2 | y = x1 closed in R2 ?
(b) connected ?
(c) compact ?
(d) homeomorphic to R2 ?
(e) discrete in the subspace topology inherited from Rl Rl ?
Justify your answ

MTG 4302/5316 Fall-2016 Pop-quiz, Name:
1. What is the cardinality of the set of functions f : N N that are
eventually zero. A function f : N N is called eventually zero if there
is n0 such that for all n > n0 , f (n) = 0.
SOLUTION: This set can be presen