CIUDAD UNIVERSITARIA DEPARTAMENTO DE MATEMÁTICAS, FACULTAD DE CIENCIAS, U.N.A.M. C.P. 04510 MÉXICO D.F. C.P. 04510 MÉXICO D.F.
UNIVERSIDAD AUTONÓMA METROPOLITANA DEPARTAMENTO DE MATEMÁTICAS AV. MICHOACAN Y LA PURIacute;SIMA IZTAPALAPA A.P. 55-532 C.P. 09340 MÉXICO D.F. AV. MICHOACAN Y LA PURIacute;SIMA IZTAPALAPA A.P. 55-532 C.P. 09340 MÉXICO D.F.
We prove that there are Tychonoff spaces X for which p(Cp(X)) =ϖ and Cp(X) is a Lindelf Σ-space while the network weight of X is uncountable. This answers Problem 75 from . An example of a space Y is given such that p(Y)=ϖ and Cp(Y) is a Lindelf Σ-space, while the network weight of Y is uncountable. This gives a negative answer to Problem 73 from . For a space X with one non-isolated point a necessary and sufficient condition in terms of the topology on X is given for Cp(X) to have countable point-finite cellularity.