Any open subset of $\\Bbb R$ is a countable union of disjoint open ...
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. …
elementary set theory - What do finite, infinite, countable, not ...
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 …
Why is it important for a manifold to have a countable basis?
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?
What does it mean for a set to be countably infinite?
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 …
Uncountable vs Countable Infinity - Mathematics Stack Exchange
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
Prove that the union of countably many countable sets is countable.
Dec 12, 2013 · So to show that the union of countably many sets is countable, we need to find a similar mapping. First, let's unpack "the union of countably many countable sets is countable": "countable …
Co-countable set and a countable set - Mathematics Stack Exchange
To be more precise, the hypothesis that X X is uncountable really comes in the statement that a set can't be both countable and co-countable. That's necessary in order for m m to even be well-defined, and …
What is the point of countable vs. uncountable sets?
4 Rather than answer countable vs uncountable specifically, I'll ramble on why Cantor's work could be considered important. AFAIK, this was the very first time anyone had made an observation of that …
Prove that the set of all algebraic numbers is countable
This is a better proof than the one suggested by the hint (which is, I believe, the proof in, say, Rudin). There are obviously infinitely many algebraic numbers (consider Q Q!), but there are at most …
elementary set theory - Why are natural numbers 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 …