A s is well known, the Stone-tech compactification of a product space is not generally identical (more precisely, homeomorphic) with the product of the Stone-tech compactifications o f coordinate spaces. M . Henriksen and J. R. Isbell [ 5 ] pointed ou t that the relation R(X x Y )= R X x R Y " implies the pseudo-compactness of the product X x Y 2 3 ) . Recently, the converse has been established by I. Glicksberg [4]. He proved more generally that the relation $ (HX )=11. i3X" holds true if and only i f HX," 4 ) is pseudo-compact.In this note, we shall restrict ourselves to consider the product o f two spaces, and give som e conditions equivalent to that theT h e pseudo-compactness o f t h e product X x Y implies the pseudo-compactness o f each coordinate space. However, it is not true that the product of pseudo-compact spaces must be pseudocompacts'. Several additional conditions sufficient to insure the pseudo-compactness of the product of pseudo-compact spaces are given and discussed in [ I ] , [4 ] and [ 5 ] . We shall generalize those results in somewhat unific form.1) Throughout, we shall consider X as a subspace o f ox.2) The trivial case that X o r Y is a finite set will be excluded throughout. If X is a finite set, th en 6 (X x Y )= 0 X x B Y for any space Y.3) T . Is h iw a ta [7 ] has proved that if /3(XX X)--fa x fa , t h e n X is totally bounded fo r any uniform structure of X . ( X is pseudo-compact if and only i f it is totally bounded for any uniform structure of X . C. f. T . Ishiwata: O n uniform spaces, Sugaku Kenkyuroku, Vol. 2 (1953) (in Japanese).) 4) II X . denotes the product of X . 5 ) C.f. [9], [10].

Abstract.This paper gives a characterization of paracompactness, and introduces the notion of an elastic space which generalizes the concept of a stratifiable (in particular, metric) space.

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Abstract. Let X be a non-paracompact subspace of a linearly ordered topological space. We prove, in particular, that if a Hausdorff topological group G contains closed copies of X and a Hausdorff compactification bX of X then G is not normal. The theorem also holds in the class of monotonically normal spaces.2000 AMS Classification: 54H11,22A05, 54F05

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