Relationships between a
topological space (more generally a convergence space) and its associated function
space C(X) are investigated. The algebra of all continuous real-valued functions on a
space X together with the continuous convergence structure is denoted by Cc(X).
After appropriate generalizations of the axioms of countability to convergence spaces,
it is shown: 1. A completely regular topological space X is Lindelöf if and
only if Cc(X) is first countable. 2. A completely regular topological space
X is separable and metrizable if and only if Cc(X) is second countable.
Generalizations of (1) and (2) are introduced, and results and examples which
justify the use of axioms of countability in convergence space theory are
presented.
