subset of dedekind infinite set is infinite

Question

I got a question: Is any subset of dedekind infinite set is infinite? or if I remove a singleton set from dedekind infinite set, is the set left infinite?

Can anyone give me an example of injective function that makes the statement true?

Answer 1

Not every subset, of course. The empty set is an example.

But even if you mean that every infinite subset of a Dedekind infinite set is Dedekind infinite as well, then if the axiom of choice holds then the answer is obviously yes because infinite sets are Dedekind infinite.

However, assuming that there are infinite Dedekind finite sets, the answer is no. Take an infinite Dedekind finite set $A$ and consider the Dedekind infinite set $A\cup\omega$, and $A$ as an infinite subset. Then we have a Dedekind infinite set with an infinite Dedekind finite subset.

Note, by the way that given any infinite set, removing a finite subset does not make the set finite all of a sudden; nor it can make it Dedekind finite.