Math 4853 homework
62. Prove that every uncountable subset of R has a limit point in R. (Let A be an uncountable subset of R, and for n Z put An = A [n, n + 1].)
63. Let cfw_ xn be a sequence in a metric space X. Prove that if xn x, then cfw_ xn is
Cauc
Math 4853 homework
26. Take as known the fact that a composition of bijections is a bijection (prove this if it
is not already clear to you).
(a) Show that if X and Y are countable sets, then there is a bijection from X to Y .
(b) Let X be a countable set
Math 4853 homework
29. (3/12) Let X be a topological space. Suppose that B is a collection of subsets of X
such that
(1) The sets in B are open in X.
(2) For every open set U in X and every x U , there exists B B such that
x B U.
(a) Prove that B is a bas
Math 4853 homework solutions (version of April 5, 2010)
18. Verify that if U is open in the standard topology on R, then it is open in the lower
limit topology on R.
Let U be open in the standard topology, and let x U . Then there exists > 0 such that
(x
Math 4853 homework solutions (version of February 23, 2010)
5. Prove directly from the denition of continuity that if f : R R and g : R R are
continuous at x0 , and g(x0 ) = 0, then the quotient function f /g is continuous at x0 .
Fix x0 and let
> 0 be gi
Math 4853 homework
9. (2/12) For 1 k n, let k : Rn R be the projection function dened by
k (r1 , . . . , rn ) = rk . Let f : Rm Rn be a function. Dene fk : Rm R by fk = k f ,
so that f (x) = (f1 (x), f2 (x), . . . , fn (x).
(a) Prove that k is continuous.
Math 4853 homework
15. (2/12) Let S(x, ) R2 be the open square of side 2 centered at x.
S(x1 , x2 ), ) = cfw_(z1 , z2 ) | |z1 x1 | < and |z2 x2 | < .
That is,
(a) Prove that S(x, ) is open. (To gure out a with B(y, ) S(x, ), draw a
picture. The argument u
Math 4853 homework
23. (3/1) Let X be a set and let B be a collection of subsets of X. Dene U = cfw_U
X | x U, B B, x B U . Prove that U U if and only if U is a union of
elements of B.
24. (3/1) Let X be a set and let B be a collection of subsets of X. W
Math 4853 homework solutions (version of February 12, 2010)
1. Let f : R R be the function dened by f (x) = x2 . Prove (directly from the
denition) that f is continuous.
Fix x0 , and suppose for now that x0 = 0. Note rst that if |x x0 | < |x0 |, then |x|
Math 4853 homework
51. (not to turn in) Let X be a set with the conite topology. Prove that every subspace
of X has the conite topology (i. e. the subspace topology on each subset A equals the
conite topology on A). Notice that this says that every subset
Math 4853 homework
36. (4/2) Prove that the continuous bijection f : [0, 2) S 1 dened by f (t) = (cos(t), sin(t)
is not a homeomorphism.
37. (4/2) (a) Let C be the conite topology on R. Prove that if f 1 (cfw_r) nite for every
r R, then f : (R, C) (R, C)
Math 4853 homework
43. (a) Let B A X. Prove that B is closed in the subspace topology on A if and only
if there exists a closed subset C X such that B = C A.
(b) Prove that if A is a closed subset of X and B is a closed subset of Y , then A B
is a closed
Math 4853 homework
35. Let f : X Y be a function between topological spaces. Let Z be a subset of Y such
that f (X) Z, and dene g : X Z by g(x) = f (x). We say that g is obtained
from f by restriction of the codomain. Assuming, of course, that Z has the s
Math 4853 homework
42. (4/2) Let X Y be a product of topological spaces. Prove that for each x0 X, the
subspace cfw_x0 Y is homeomorphic to Y . Show, in fact, that the restriction of the
projection function Y : X Y Y is a homeomorphism. Hint: Let j : Y c
Math 4853 homework
1. (due 2/3) Let f : R R be the function dened by f (x) = x2 . Prove (directly from
the denition) that f is continuous.
2. (2/3) Let f : R R be the function dened as follows.
(q 1)/q x is rational and x = p in lowest terms with p 0 and