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 …
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. …
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
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 …
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 …
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?
How to prove that $\\mathbb{Q}$ ( the rationals) is a countable set
Since the Cartesian product of two countable sets is countable (see for example the wiki article Pairing function), if [0, 1)Q [0, 1) Q is countable then so is Q Q. We can build a 1: 1 1: 1 correspondence …
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?
In terms of the applied and for practical purposes for real analysis: additivity of measure is countable. And, much work is done in formalizing analysis in the constructible and countable. It's good to know …
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 …