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 …
Idea of proving that a countable union of countable sets is countable ...
Aug 7, 2025 · 2 There were some previous discussions and the consensus was that AC (or ACC, axiom of countable choice) is required to prove the fact that a countable union of countable sets 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 …
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?
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
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 …
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 …
Second Countable, First Countable, and Separable Spaces
Mar 16, 2015 · Thus each second countable space is first countable. Now if the space X is second-countable, to also be separable, there needs to exists a countable dense subset of X.
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. …
If $S$ is an infinite $\\sigma$ algebra on $X$ then $S$ is not countable
Once you have a infinite collection of pairwise disjoint sets one can identify each of these as distinct elements where unions of sets are also distinct. So by taking all countable unions on this collection …