Question

Let the set N of natural numbers be endowed with the cofinite topology (in which a set is open if and only if it is empty or its complement is finite). (a) Is N connected? Justify your answer. (b) Is N compact? Justify your answer. (c) Explain why the function f : N → N, n→ n ^3 is continuous. (d) Exhibit a function g : N → N which is not continuous.

Answer #1

Real Topology: let A={1/n : n is natural} be a subset of the real
numbers. Is A open closed, or neither? Justify your answer.

Let X be a topological space with topology T = P(X). Prove that
X is finite if and only if X is compact. (Note: You may assume you
proved that if ∣X∣ = n, then ∣P(X)∣ = 2 n in homework 2, problem 2
and simply reference this. Hint: Ô⇒ follows from the fact that if X
is finite, T is also finite (why?). Therefore every open cover is
already finite. For the reverse direction, consider the
contrapositive. Suppose X...

Let S be the set {(-1)^n +1 - (1/n): all n are natural
numbers}.
1. find the infimum and the supremum of S, and prove that these
are indeed the infimum and supremum.
2. find all the boundary points of the set S. Prove that each of
these numbers is a boundary point.
3. Is the set S closed? Compact? give reasons.
4. Complete the sentence: Any nonempty compact set has a....

Let T be the half-open interval topology for R, defined in
Exercise 4.6.
Show that (R,T) is a T4 - space.
Exercise 4.6
The intersection of two half-open intervals of the form [a,b) is
either empty or a half-open interval. Thus the family of all unions
of half-open intervals together with the empty set is closed under
finite intersections, hence forms a topology, which has the
half-open intervals as a base.

Using Discrete Math
Let ρ be the relation on the set of natural numbers N given by:
for all x, y ∈ N, xρy if and only if x + y is even. Show that ρ is
an equivalence relation and determine the equivalence classes.

If we let N stand for the set of all natural numbers, then we
write 6N for the set of natural numbers all multiplied by 6 (so 6N
= {6, 12, 18, 24, . . . }). Show that the sets N and 6N have the
same cardinality by describing an explicit one-to-one
correspondence between the two sets.

let F : R to R be a continuous function
a) prove that the set {x in R:, f(x)>4} is open
b) prove the set {f(x), 1<x<=5} is connected
c) give an example of a function F that {x in r, f(x)>4} is
disconnected

Let S = {0,1,2,3,4,...}, A = the set of natural numbers
divisible by 2, and B = the set of numbers divisible by 5. What is
the set A intersection B? What is the set A union B? Please show
your work.

Exercise 6.6. Let the inductive set be equal to all natural
numbers, N. Prove the following propositions. (a) ∀n, 2n ≥ 1 +
n.
(b) ∀n, 4n − 1 is divisible by 3.
(c) ∀n, 3n ≥ 1 + 2 n.
(d) ∀n, 21 + 2 2 + ⋯ + 2 n = 2 n+1 − 2.

Let N2K be the set of the first 2k natural numbers.
Prove that if we choose k + 1 numbers out of these 2k, there is at
least one pair of numbers a, b for which a is divisible by b.

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