Let {Kn : n ∈ N} be a collection of nonempty compact subsets of
R N...
Let {Kn : n ∈ N} be a collection of nonempty compact subsets of
R N such that for all n, Kn+1 ⊂ Kn. Show that K = T∞ n=1 Kn is
compact. Can K ever be the empty set?
Let S ⊆ R be a nonempty compact set and p ∈ R. Prove that there
exists a point x_0 ∈ S which is “closest” to p. That is, prove that
there exists x0 ∈ S such that |x_0 − p| is minimal.
Let A ⊂ R be a nonempty discrete set
a. Show that A is at most countable
b. Let f: A →R be any function, and let p ∈ A be any point. Show
that f is continuous at p
Let Ω be any set and let F be the collection of all subsets of Ω
that are either countable or have a countable complement. (Recall
that a set is countable if it is either finite or can be placed in
one-to-one correspondence with the natural numbers N = {1, 2, . .
.}.)
(a) Show that F is a σ-algebra.
(b) Show that the set function given by
μ(E)= 0 if E is countable ;
μ(E) = ∞ otherwise...
Let S be a set of n numbers. Let X be
the set of all subsets of S of size k, and let
Y be the set of all ordered k-tuples
(s1, s2, ,
sk)
such that
s1 < s2
< < sk.
That is,
X
=
{{s1, s2, ,
sk} | si S and all si's
are distinct}, and
Y
=
{(s1, s2, ,
sk) | si S and s1 <
s2 < < sk}.
(a) Define a one-to-one correspondence
f : X → Y.
Explain...
Let S(n) be the number of subsets of {1,2,...,n} having the
following property: there are no three elements in the subset that
are consecutive integers. Find a recurrence for S(n) and explain in
words why S(n) satisfies this recurrence
Let (E,d) be a metric space and K, K' disjoint compact subsets
of E. Prove the existence of disjoint open sets U and U' containing
K and K' respectively.
Definition 1 (Topological space). Let X be a set. A collection O
of subsets of X is called a topology on the set X if the following
properties are satisfied:
(1) emptyset ∈ O and X ∈ O.
(2) For all A,B ∈ O, we have A∩B ∈ O (stability under
intersection).
(3) For all index sets I, and for all collections {Ui}i∈I of
elements of O (i.e., Ui ∈ O for all i ∈ I), we have U i∈I...
Let A and B be two non empty bounded subsets of R:
1) Let A +B = { x+y/ x ∈ A and y ∈ B} show that sup(A+B)= sup A
+ sup B
2) For c ≥ 0, let cA= { cx /x ∈ A} show that sup cA = c sup
A
hint:( show c supA is a U.B for cA and show if l < csupA then
l is not U.B)
Let R be a ring and n ∈ N. Let S = Mn(R) be the ring of n × n
matrices with entries in R.
a) i) Let T be the subset of S consisting of the n × n diagonal
matrices with entries in R (so that T consists of the matrices in S
whose entries off the leading diagonal are zero). Show that T is a
subring of S. We denote the ring T by Dn(R).
ii). Show...