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In this paper, on the family (Y) of all open subsets of a space Y we define the so called quasi Scott topology, denoted by ${\tau }_{qSc}$. This topology defines in a standard way, on the set C(Y,Z) of all continuous maps of the space Y to a space Z, a topology $t_{qIs}$ called the quasi Isbell topology. The latter topology is always larger than or equal to the Isbell topology, and smaller than or equal to the strong Isbell topology. Results and problems concerning the topology $t_{qIs}$ are given.

In questo articolo vengono presentate e studiate le nozioni di insieme ${\mathrm{\Lambda }}_{\delta }$ e di insieme $\left(\mathrm{\Lambda },\delta \right)$-chiuso. Inoltre, vengono introdotte le nozioni di $\left(\mathrm{\Lambda },\delta \right)$-continuità, $\left(\mathrm{\Lambda },\delta \right)$-compatezza e $\left(\mathrm{\Lambda },\delta \right)$-connessione e vengono fornite alcune caratterizzazioni degli spazi $\delta -T_0$ e $\delta -T_1$. Infine, viene mostrato che gli spazi $\left(\mathrm{\Lambda },\delta \right)$-connessi e $\left(\mathrm{\Lambda },\delta \right)$-compatti vengono preservati mediante suriezioni $\delta$-continue.