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 …

Nov 14, 2021 · In second-countable spaces—as in metric spaces—compactness, sequential compactness, and countable compactness are all equivalent properties. Question: So …

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.

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 …

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Apr 29, 2017 · Munkres in his book states that: Theorem 30.3 Suppose that X has countable basis, then every open covering of X contains a countable subcollection covering X. Proof. Let …

Feb 22, 2016 · I using the definition - If a set is countably infinite, then each element of the set can be mapped to the set of natural numbers. Another question - Is the cardinal product of …

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 …

Sep 26, 2021 · If a space is second-countable, the usual way to prove this will be to exhibit a specific countable basis. If a space is not second-countable, proving this can be more intricate, …

Feb 6, 2015 · From the fact that cartesian product of countably infinite sets is countably infinite, $\mathbb Z \times \mathbb N$ is countable. Because the domain of injection to a countable …