Question

Prove that any countable subset of [a,b] has measure zero. Recall that a set S has measure zero if there is a countable collection of open intervals with .

Answer #1

[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.

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

why
is every countable subset a zero set? real analysis

if A subset B is open then A is the countable union of bounded
open intervals

Verify: any countable ordered set is similar to a subset of Q
intersect (0,1).

Prove : If S is an infinite set then it has a subset A which is
not equal to S, but such that A ∼ S.

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.

true or false?
every uncountable set has a countable subset. explain

prove that the lebesgue measure on R has a countable
basis

Prove for each:
a. Proposition: If A is finite and B is countable, then A ∪ B is
countable.
b. Proposition: Every subset A ⊆ N is finite or countable.
[Similarly if A ⊆ B with B countable.]
c. Proposition: If N → A is a surjection, then A is finite or
countable. [Or if countable B → A surjection.]

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