Increasing chain of countable sets
= Increasing chain of countable sets
{title2=$X=\bigcup_{i\in I}X_i$}
= Linear increasing union of countable sets
{synonym}
A family of countable <sets> indexed by a <linear order> is increasing if $i\leq j$ implies $X_i\subseteq X_j$. Its union has <cardinality> at most $\aleph_1$, by the <cardinal bound for an increasing chain of countable sets>. The index order need not be a <well-order>.