Question

Show that every infinite semi-decidable language A has an infinite subset B⊆A such that B is a decidable language.

Answer #1

Proposition (Bolzano’s theorem). Every bounded infinite subset of R
has at least one accumulation point.

Show that every nonempty subset of the real numbers with a lower
bound has a greatest lower bound.

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

[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 a subset of a countably infinite set is finite or
countably infinite.

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

Must every linearly dependent set have a subset that is
dependent and a subset that is independent?

Let [a],[b],[c] be a subset of Zn. Show that if [a]+[b]=[a]+[c],
then [b]=[c].

Suppose that E is a closed connected infinite subset of a metric
space X. Prove that E is a perfect set.

(2)
If K is a subset of (X,d), show that K is compact if and only if
every cover of K by relatively open subsets of K has a finite
subcover.

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