Recently I've been reading "Principles of Mathematical Analysis" by Rudin, and have just begun the section on basic topology. The first theorem it presents is the following:
Theorem Every infinite subset of a countable set A is countable.
It then provides a proof, and at the end, generalizes by saying that, roughly speaking, countable sets represent the "smallest" infinity: No uncountable set can be a subset of a countable set.
Now, from my current understanding, a set being countable or not depends on if there is an equivalence relation between the set in question, and the set of all positive integers.
I'm wondering if this is similiar to an upper bound property..
Ex. Set A is defined on the interval [0,10] consisting of all natural numbers within that interval. It is finite, and countable. Now, take the [0,10] and define set B to be all rational numbers within the interval. It is infinite, but still countable, since set A is countable.
Is this what the theorem is stating, in essence? Also, could someone please give an example of where this is not true?