In recent times, countable athleticallyrelated activities has become increasingly relevant in various contexts. elementary set theory - What do finite, infinite, countable, not .... 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
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. Moreover, so, the set $\Bbb Q$ is countable in spite of being infinite. Second Countable, First Countable, and Separable Spaces.
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. general topology - Proof that Metric Spaces are Second Countable .... In second-countable spaces—as in metric spaces—compactness, sequential compactness, and countable compactness are all equivalent properties. Question: So apparently, metric spaces are always second countable; however, how can we prove this?
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 general topology - How to tell if a space is second-countable .... If a space is second-countable, the usual way to prove this will be to exhibit a specific countable basis.
Another key aspect involves, if a space is not second-countable, proving this can be more intricate, and what methods are promising will vary a lot with the kind of space you are looking at. Any second-countable space is separable, and disproving separability can be ... Why is $\mathbb Q $ (rational numbers) countable? From the fact that cartesian product of countably infinite sets is countably infinite, $\mathbb Z \times \mathbb N$ is countable.
In this context, because the domain of injection to a countable set is countable, $\mathbb Q$ is countable. elementary set theory - Intersection of countable and uncountable sets .... yess it seems that here "countable"="countably infinite" and "at most countable"="empty or finite or countably infinite". In next question (quiz), it says intersection of finite and uncountable set is "finite" but not countable. This perspective suggests that, any open subset of $\\Bbb R$ is a countable union of disjoint open ....
Equally important, 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?
Similarly, the equivalence between paracompactness and second countablity in a .... you probably do not mean that each component is countable, but that the space has countably many components.
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