C-COMPACT AND FUNCTIONALLY COMPACT SPACES

In the first section of this note a question posed by G. Viglino is resolved by constructing a C-compact space which is not seminormal. In the second section some characteriza-tions of C-compact and functionally compact spaces are introduced. In the final section, embedding theorems of spaces into C-compact and functionally compact spaces are noted. 1 * A Counterexample* DEFINITIONS, (a) A Hausdorff space X is absolutely closed if given an open cover 7 " of X, then there exists a finite number of elements of % " say Vi9 l^i^n, with X c Cl JJ?=i V,. (b) A Hausdorff space (X, r) is C-compact if given a closed set Q of X and a τ-open cover °F of Q, then there exists a finite number of elements of % " say Vi9 1 ^ i ^ n, with Q c Clx JJ?=ι Vt. (c) An open set V is regular if V = F°. (d) A space X is seminormal if given a closed subset C of I and an open set V containing G, then there exists a regular open set R with C c J J c F. G. Viglino has shown that a seminormal absolutely closed space is C-compact, and posed the question as to whether or not the converse holds [5]. The following is an example of a C-compact space which is not seminormal. An example has also been obtained by T. Lominac, Abstract #682-54-33. EXAMPLE. Let Z represent the set of positive integers. Le