  We consider the class of interval maps with dense set of periodic points CP
and its closure Cl(CP) equipped with the metric of uniform convergence. Besides
studying basic topological properties and density results in the spaces CP and
Cl(CP) we prove that Cl(CP) is dynamically characterized as the set of interval
maps for which every point is chain-recurrent. Furthermore, we prove that a
strong topological expansion property called topological exactness (or leo
property) is attained on the open dense set of maps in CP and on a residual set
in Cl(CP). Moreover, we show that every second category set in CP and Cl(CP) is
rich in a sense that it contains uncountably many conjugacy classes. An
analogous conclusion also holds in the setting of interval maps preserving any
fixed non-atomic probability measure with full support. Finally, we give a
detailed description of the structure of periodic points of generic maps in CP
and Cl(CP) and show that generic maps in CP and Cl(CP) satisfy the shadowing
property.
