Given a set $\Sigma$ of letters, apply the Kleene star operation to it, and we get $\Sigma^*$, the set of all finite-length sequences from $\Sigma$, called strings (allowing a letter appearing more than once in a string).
If $\Sigma$ is empty, then $\Sigma^*$ consists only one string, the empty string.
If $\Sigma$ is not empty, then $\Sigma^*$ has an infinite cardinality.
I wonder if $\Sigma^*$ can be countably infinite? When will that be true? When $\Sigma$ is finite?
When will its cardinality is uncountably infinite?
Thanks!
$\sum^{\star} = \bigcup_{k=0}^{\infty} \sum^{k}$, where $\sum^{k}$ is a set of strings of letters from $\sum$ which have a length is equals to k.
So if alphabet $\sum$ is finite or countable then $\sum^{\star}$ is countable.
On the other hand if a cardinality of $\sum^{\star} = \bigcup_{k=0}^{\infty}(\sum^{k}\setminus\sum^{k-1})$, let $\sum^{-1}=\emptyset$, is uncountable then some of $\sum^{k}\setminus\sum^{k-1}\sim \sum\times...\times\sum$ (k times) must be uncountable, hence $\sum$ must be uncountable.