Iterated function spaces and Lindelöf \Sigma-property
by
Vladimir V. Tkachuk
Universidad Autonoma Metropolitana de Mexico

We prove that if C_p(X) is a Lindelöf \Sigma-space, then so is C_p(\nuX), where \nuX is the Hewitt realcompactification of the space X. One of the important consequences is that if C_p(X) is a Lindelöf \Sigma-space, then C_{p, n}(X) is a Lindelöf \Sigma-space for every odd n. Another consequence is that if the second C_p is a Lindelöf \Sigma-space, then so are all even C_p's.

The results in question show that there are only four possibilites for the distribution of the Lindelöf \Sigma-property in iterated function spaces:

1) no iterated function space is a Lindelöf \Sigma-space;

2) all iterated function spaces are Lindelöf \Sigma-spaces;

3) only even iterated function spaces are Lindelöf \Sigma-spaces;

4) only odd iterated function spaces are Lindelöf \Sigma-spaces;

Date received: May 17, 1999