Quasi Component Research Articles

δµ -connectedness in a µ-proximity space

Abstract In this paper we introduce the notion of δµ -connectedness on a µ-proximity space. It has been proved that δµ -connectedness can be characterized by δµ -continuous functions. We initiate the idea of δµ -chain and establish some results related to this. The concepts of δµ -component and δµ -quasi component have been introduced and their interrelation has been studied.

Compactness, connectedness and Countability properties of C(X) with the r-topology

We show that the component and quasi component of 0 in the space C r (X) (C(X) with r-topology), coincide with the intersection of all principal ideals in C*(X) generated by regular elements of C*(X). Several different characterizations of the component of 0 in C r (X) are given and using these characterizations, it turns out that C r (X) is totally disconnected if and only if the set of non-almost P-points of β X is dense in β X. It is also shown that C r (X) is connected or locally connected if and only if X is a pseudocompact almost P-space. We observe that most of the familiar forms of compactness and most of the countability properties of C r (X) are equivalent to the finiteness of X. Finally, open and closed ideals in C r (X) are investigated and it is proved that for every \({p \in X}\), cl r O p = M p if and only if X is an almost P-space.