What does it mean for a set to be countably infinite? If you can achieve a bijection of the members of the sets to $\Bbb N$, the the set will be called countable, and moreover ,if it is infinite, then it is countably infinite So, the set $\Bbb Q$ is countable in spite of being infinite
Uncountable vs Countable Infinity - Mathematics Stack Exchange My friend and I were discussing infinity and stuff about it and ran into some disagreements regarding countable and uncountable infinity As far as I understand, the list of all natural numbers is
Co-countable set and a countable set - Mathematics Stack Exchange You'll need to complete a few actions and gain 15 reputation points before being able to upvote Upvoting indicates when questions and answers are useful What's reputation and how do I get it? Instead, you can save this post to reference later
Any open subset of $\\Bbb R$ is a countable union of disjoint open . . . 9 $\mathbb {R}$ with standard topology is second-countable space For a second-countable space with a (not necessarily countable) base, any open set can be written as a countable union of basic open set Given any base for a second countable space, is every open set the countable union of basic open sets?
How to prove that the set of rational numbers are countable? Any set that can be put in one-to-one correspondence in this way with the natural numbers is called countable In some sense, this means there is a way to label each element of the set with a distinct natural number, and all natural numbers label some element of the set
Why is it important for a manifold to have a countable basis? I would like to understand the reason why we ask, in the definition of a manifold, for the existence of a countable basis Does anybody have an example of what can go wrong with an uncountable basis?
Prove that the set of all algebraic numbers is countable This is a better proof than the one suggested by the hint (which is, I believe, the proof in, say, Rudin) There are obviously infinitely many algebraic numbers (consider $\mathbb {Q}$!), but there are at most countably many of them since there are only countably many coefficients, each of these contributing finitely many roots