Question

Prove or provide a counterexample

If A is a nonempty countable set, then A is closed in T_H.

Answer #1

41. Prove that a proper subset of a countable set is
countable

Prove that a countable union of countable sets countable; i.e.,
if {Ai}i∈I is a collection of sets, indexed by I ⊂ N, with each Ai
countable, then union i∈I Ai is countable. Hints: (i) Show that it
suffices to prove this for the case in which I = N and, for every i
∈ N, the set Ai is nonempty. (ii) In the case above, a result
proven in class shows that for each i ∈ N there is a...

Use the fact that “countable union of disjoint countable sets is
countable" to prove “the set of all polynomials with rational
coefficients must be countable.”

Prove that the set of constructible numbers is countable

Let A be a nonempty set. Prove that the set S(A) = {f : A → A |
f is one-to-one and onto } is a group under the operation of
function composition.

Prove:
A nonempty subset C⊆R is closed if and only if
there is a continuous function g:R→R such that
C=g-1(0).

Prove the conjecture or provide a counterexample:
Let U ∈ in the usual topology, and let F be a finite set. Then
(U−F) ∈ in the usual topology.

Prove whether or not the set ? is countable.
a. ? = [0, 0.001)
b. ? = ℚ x ℚ
I do not really understand how to prove
S is countable.

[Q] Prove or disprove:
a)every subset of an uncountable set is countable.
b)every subset of a countable set is countable.
c)every superset of a countable set is countable.

Prove that the set of all finite subsets of Q is countable

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