= Cardinality of continuous maps determined on a countable dense set
{title2=$|C(X,\mathbb R)|=2^{\aleph_0}\quad(X\ne\varnothing\text{ separable})$}
Continuous <real-valued functions> on a space with a <countable> <dense subset> are determined by their restrictions to that <subset>. The equality <set> of two such <functions> is closed and contains the <dense subset>. Hence there are at most $|\mathbb R|^{\aleph_0}=2^{\aleph_0}$ such <functions>. If the domain is nonempty, distinct <constant functions> give the matching <lower bound>.
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