The paper, consisting of four sections, of which Section 0 is auxiliary, is devoted to some principal questions of the theory of bitoplogical spaces. In Section 1, the p-extremally disconnected, (i,j)-strongly extremally disconnected and (i,j)-nodec spaces are studied by means of the localization at a point. In Section 2, the (i,j)-pseudoscattered, (i,j)-nd-scattered, p-ultradisconnected and (i,j)-Moscow spaces are introduced, their interrelations and their relations with the p-extremally disconnected spaces are investigated, in particular, when one of the topologies is finer than the other. Section 3 is concerned with the problem of existence of a finer connected topology on a connected topological space and so, it is related to the concept of a maximal connected space.
C-, N- and D-relations; (i,j)-Baire space in the strong sense, WS-tangency and weak tangency of topologies, (i,j)-strongly disconnected, p-ultradisconnected, p-crowded, (i,j)-nodec, (i,j)-nd-scattered, (i,j)-pseudoscattered and (i,j)-Moscow spaces, point of i,j)-(strong)-extremal disconnectedness, (i,j)-nodec and (i,j)-P-points.
MSC 2000: 54E55