Spaces in which each
regular closed subset is an intersection of a sequence of closed neighborhoods
are investigated. This property is shown to be equivalent to each of the
following: Each regular closed subset is a zero set of a continuous function.
Each normal lower semicontinuous function defined on the space is a limit
of an increasing sequence of continuous functions. The space satisfies the
strong C insertion property for normal semicontinuous functions. Separation
properties of X × I, which are weaker than normality, are related to the
insertion of a continuous function between two comparable functions defined on
X.
