Some notes on densely continuous forms

Abstract

Abstract We consider a special space of set-valued functions (multifunctions), the space of densely continuous forms D(X, Y) between Hausdorff spaces X and Y, defined in [HAMMER, S. T.—McCOY, R. A.: Spaces of densely continuous forms, Set-Valued Anal. 5 (1997), 247–266] and investigated also in [HOLÁ, L’.: Spaces of densely continuous forms, USCO and minimal USCO maps, Set-Valued Anal. 11 (2003), 133–151]. We show some of its properties, completing the results from the papers [HOLÝ, D.—VADOVIČ, P.: Densely continuous forms, pointwise topology and cardinal functions, Czechoslovak Math. J. 58(133) (2008), 79–92] and [HOLÝ, D.—VADOVIČ, P.: Hausdorff graph topology, proximal graph topology and the uniform topology for densely continuous forms and minimal USCO maps, Acta Math. Hungar. 116 (2007), 133–144], in particular concerning the structure of the space of real-valued locally bounded densely continuous forms D p*(X) equipped with the topology of pointwise convergence in the product space of all nonempty-compact-valued multifunctions. The paper also contains a comparison of cardinal functions on D p*(X) and on real-valued continuous functions C p(X) and a generalization of a sufficient condition for the countable cellularity of D p*(X).