"Countably infinite" is unambiguous, but some authors use "countable" to mean countably infinite, while many (perhaps most) use countable to mean finite or countably infinite, as the other answers indicate. When authors use countable to refer only to sets in bijection with $\mathbb{N}$, they often end up using the phrase "at most countable ...

Clearly every finite set is countable, but also some infinite sets are countable. Note that some places define countable as infinite and the above definition. In such cases we say that finite sets are "at most countable".

2018年6月24日 · There are many text that define countable and finite as mutually incompatible. A finite set is not countable and a countable set must be infinite. Many others define countable as including finite as well and countably infinite. BOTH conventions will define "at most countable" as "not uncountable" to include the options of being finite or empty.

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2019年2月6日 · Cartesian product of countable set A. 10. Induction - Countable Union of Countable Sets. 1.

2018年12月10日 · To demonstrate that ANY countable set has measure zero, is it sufficient to show that the natural numbers have a measure zero? If so, why; and, if not, why not? Thank you :)

2020年6月2日 · Every at-most-countable set is closed in the sense of $\mathcal{T}$, but the "in the sense of $\mathcal{T}$ "-bit is often dropped when there is only one topology being considered. Remember that the closed sets are by definition the complements of the open sets.

You are right with the injection you gave; any finite set is trivially countable, hence you can use the same set for part 1 and 2. When we are talking about infinite countable sets (eg $\mathbb{N}$) we use the terminology countably infinite (in which case, as shown by cantor, there does not exist an injection from the power set to $\mathbb{N ...

Problem statement. Prove that if $A$ is an uncountable set and $B$ is a countable set, then $A\setminus B$ must be uncountable.

2015年9月20日 · A countable set contained in $[0,1]$ with no limit points. The question asks if such a request is possible. I think it is. Consider $\{1/n\}$. This is a countable set and it's a subset of $[0,1]$. If I wanted a countable set in [0,1] with no isolated points, this would be impossible I …