Countable Infinite And Uncountable Infinite

In recent times, countable infinite and uncountable infinite 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 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? 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. real analysis - Open sets can be expressed as a countable, disjoint ....

The fact that it can be expressed as a countable, disjoint union, of some intervals is known and has been dealt with on SE in multiple posts like in here or here just to name a couple. Show that the set of all finite subsets of $\mathbb {N}$ is countable.. Similarly, but, the simplest way to see that the set of all finite subsets of $\mathbb {N}$ is countable is probably the following. If you can list out the elements of a set, with one coming first, then the next, and so on, then that shows the set is countable. There is an easy pattern to see here.

Equally important, just start out with the least elements. Any open subset of $\\Bbb R$ is a countable union of disjoint open .... 9 $\mathbb {R}$ with standard topology is second-countable space. Moreover, 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?

Equally important, every separable metric space has a countable base. Prove that every separable metric space has a countable base. Proof: Let (X,d) (X, d) be a metric space and D = {dj}j∈N D = {d j} j ∈ N be the countable and dense subset of X X.

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. 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 ...

This perspective suggests that, a metric space is separable iff it is second countable. Conversely, if you have a countable dense set, consider the family of all open balls with rational radius centered in your set. This family is countable, and it is easy to show that it is a base.