SEQUENTIAL ORDER OF HOMOGENEOUS AND PRODUCT SPACES

In this paper we will present some examples of sequen tial spaces with properties related to their sequential order [1]. In section 1 we will show that homogeneous spaces may have any p~escribed sequential order. In sec tion 2 we will show that the sequential order of the sequen tial coreflection of a product of spaces of "small " sequen tial order may have "large " sequential order and will answer a question posed by Michael [4] when we give an example of a sequential HO-space Z so that the sequential coreflection of z2 is not regular. Our constructions use sets of sequences as underlying sets and will be facili tated by the following notation. If X is a set, we denote finite sequences in X by (xo,xl,···,xk),where k E wand every x E X. If {xo'xl,···,xk} C X and SeX, we letj (xo'xl,···,xk,S) = {(xo,xl,···,xk,xk+t: x k +l E S} and le