Domination by metric spaces

Following the definition of domination of a topological space X by a metric space M introduced by Cascales, Orihuela and Tkachuk (2011) in [3], we define a topological cardinal invariant called the metric domination index of a topological space X as minimum of the set { w ( M ) : M is a metric space that dominates X } . This invariant quantifies or measures the concept of M-domination of Cascales et al. (2011) [3]. We prove (in ZFC) that if K is a compact space such that C p ( K ) is strongly dominated by a second countable space then K is countable. This answers a question by the authors of Cascales et al. (2011) [3].