On the multiplicativity of certain properties of spaces of mappings in the topology of pointwise convergence (Q1065418)

Two properties are studied for powers of the space \(C_ p(X)\) of continuous real-valued functions on X with the topology of pointwise convergence. The point-finite cellularity of a space X is defined by \(p(X)=\sup \{(| \gamma |:\gamma\) is a point-finite family of nonempty open subsets of \(X\}\). Also a space X is a Fréchet-Urysohn space provided that whenever x is in the closure of a subset A of X, then there is a sequence in A converging to x in X. The author shows that whenever \(C_ p(X)\) is a Fréchet-Urysohn space, then so is \((C_ p(X))^{\aleph_ 0}\). Also \(p((C_ p(X))^ c)=p(C_ p(X))\) for \(c=2^{\aleph_ 0}\). In particular, for any power \(\tau\), \((C_ p(X))^{\tau}\) has countable point-finite cellularity whenever \(C_ p(X)\) does.