What do finite, infinite, countable, not countable, countably infinite mean? [duplicate] Ask Question Asked 13 years, 2 months ago Modified 13 years, 2 months ago

Nov 25, 2015 · 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 …

Mar 16, 2015 · The definition for 2nd countable topological space is that "the topological space has a countable basis." My confusion is whether or not they mean countably infinite basis only.

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 …

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

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 …

you probably do not mean that each component is countable, but that the space has countably many components. If so, please edit to say "has countably many components". Show that the …

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, …

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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 …