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?

Sep 29, 2023 · On Encyclopedia of Mathematica I find the definition of the two axioms of countability in topology. [First][1] axiom: (...)A topological space satisfies the first axiom of countability if the defin...

Feb 5, 2014 · It depends on how countability is defined. I have seen "countable" referring to infinite sets only, and in cases where countability or finite cardinality needs to be shown, the problems typically …

No subset of the real numbers is countable. True or false. In looking at the wikipedia article for real numbers, I'm not really clear on the answer to this. They use the words computable and count...

Dec 10, 2016 · 1 In the mathematical literature, discrete mathematics has been characterized as the branch of mathematics dealing with "Countable Sets". On the other hand, it is well-known that …

How to define countability of $\omega^ {\omega}$ and $\omega_1$? in set theory? Ask Question Asked 13 years, 3 months ago Modified 5 years, 6 months ago

Rudin Ex 2.2: Proving countability of algebraic numbers with a hint Ask Question Asked 10 years, 10 months ago Modified 3 years, 5 months ago

This is due to the uncountable nature of the reals. So Cantor's ideas of uncountability and countability give us a precise way to think about just how manageable, or ridiculously out-of-hand, the size of an …

Still today countability is almost always defined in Cantor's words. Are the natural numbers really necessary to define countability. Most mathematicians admit that set theory is still a rich subject to …

What is a countable set? In Wikipedia we read this definition: In mathematics, a countable set is a set with the same cardinality (number of elements) as some subset of the set of natural number...