A set A A is infinite, if it is not finite. The term countable is somewhat ambiguous. (1) I would say that countable and countably infinite are the same. That is, a set A A is countable (countably infinite) if …

Nov 5, 2015 · 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

Nov 25, 2015 · If you can achieve a bijection of the members of the sets to N N, the the set will be called countable, and moreover ,if it is infinite, then it is countably infinite. So, the set Q Q is countable in …

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?

Mar 16, 2015 · Thus each second countable space is first countable. Now if the space X X is second-countable, to also be separable, there needs to exists a countable dense subset of X X.

Note that you cannot have two (or more) disjoint co-countable sets.

Aug 7, 2025 · 2 There were some previous discussions and the consensus was that AC (or ACC, axiom of countable choice) is required to prove the fact that a countable union of countable sets is …

9 R 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. …

Sep 23, 2014 · Now I know that the interval [0,1) [0, 1) for example is uncountable so I don't understand how the closed interval [0,1] [0, 1] is countable. So, how can a set of them be countable?

Sep 6, 2015 · I get how Cantor's diagonalization argument works for real numbers, but I don't see why you can't apply the same logic to natural numbers. I was reading this thread, but the explanations …