Some notes on densely continuous forms

Year, pages: 2008, 509 - 520

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.: \textit{Spaces of densely continuous forms}, \mbox{Set-Valued Anal. \textbf{5}} (1997), 247–266] and investigated also in [HOLÁ, \mL.: \textit{Spaces of densely continuous forms, USCO and minimal USCO maps}, Set-Valued Anal. \textbf{11} (2003), 133–151]. We show some of its properties, completing the results from the papers [HOLÝ, D.—VADOVIČ, P.: \textit{Densely continuous forms, pointwise topology and cardinal functions}, Czechoslovak Math. J. \textbf{58(133)} (2008), 79–92] and [HOLÝ, D.—VADOVIČ, P.: \textit{Hausdorff graph to po lo gy, proximal graph topology and the uniform topology for densely continuous forms and minimal USCO maps}, Acta Math. Hungar. \textbf{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 $D^*_p(X)$.

Vadovič, P. 2008. Some notes on densely continuous forms. In Mathematica Slovaca, vol. 58, no.4, pp. 509-520. 0139-9918.

Vadovič, P. (2008). Some notes on densely continuous forms. Mathematica Slovaca, 58(4), 509-520. 0139-9918.