A space $X$ has the DFCC (discrete finite chain condition ) provided every discrete family of nonempty open sets is finite. (see A survey on star covering propertiesMV Matveev - Topology Atlas, preprint, 1998)
Does the discrete family of nonempty open sets mean that every point $x \in X $ has a nbhd that intersects at most one set of the given family?
However, in the proof of the Theorem 14 in the charpter 2, why he said if the sequence $S=\{U_n:n\in \omega\}$ is not DFCC, then there exists $x \in X$ is an accumulation point for $S$, each nbhd of $x$ meets infinitely many $U_n$'s? Why is "infinitely"? This is puzzling me.
Thanks for any help:)