In this paper, we show that for generalized ordered spaces, para-compactness is equivalent to Property D, where a space X is said to have Property D if, given any collection {G(x): x ∈ X} of open sets in X satisfying x ∈ G(x) for each x, there is a closed discrete subset D of X satisfying X = {G(x): x ∈ D}.

Introduction. Kwun and Szczarba [3.] established a sum theorem for Whitehead torsion of homotopy equivalence. Their theorem is stated as follows.Sum Theorem. Let f" X--.Y be the sum of cellular maps fl" X1 YI and f" X-Y., where X= XI U X2, and Y Y [3 Y. are finite cell complexes. Suppose tha$ Xo-X X is 1-connected, and that f , f and fo-f Xo (= f Xo) are homotopy equivalences. Then f is a homotopy equivalence andRecently, Hosokawa [1] has generalized this theorem to the case of X0 being non-simply connected. Indeed, for this general case he obtained the following equality r(f) ],r(f) + ]=.r( f =) ]o. r( f o) where ]0." Wh (zrY0)-Wh (Y). His proof is accomplished by a geometric idea.The purpose of this paper is to extend the definition of Whitehead torsion and to prove the generalized sum theorem by making use of the torsions in the extended sense.For details on the notions of Whitehead group and torsion, we refer the reader to Whitehead [5] and Milnor [4].1. Let (K, L) be a pair consisting of a finite, connected CWcomplex K, and a subcomplex L which is a deformation retract of K.

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Abstract. The notion of the S δ -diagonal was introduced by H. R.Bennett to study the quasi-developability of linearly ordered spaces. In an earlier paper, we obtained a characterization of topological spaces with an S δ -diagonal and we showed that the S δ -diagonal property is stronger than the quasi-G δ -diagonal property. In this paper, we define a dense S δ -diagonal of a space and show that two linearly ordered extensions of a generalized ordered space X have dense S δ -diagonals if the sets of right and left looking points are countable.Keywords: S δ -diagonal, dense S δ -diagonal, linearly ordered space (LOTS), generalized ordered space (GO-space), linearly ordered extension.2000 AMS Classification: 54F05. S δ -diagonalsWe review in this section the definitions of S δ -set and S δ -diagonal, and state our results obtained in [5].The following definition is a generalization of a G δ -set and was introduced by H. R. Bennett [2] to study the quasi-developability of linearly ordered (topological) spaces. Definition 1.1. Let X be a topological space. A subset A of X is called an S δ -set if there exists a countable collection {U (1), U (2), . . .} of open subsets of X such that, for two points p ∈ A and q ∈ X \ A, there exists an n such that p ∈ U (n) and q / ∈ U (n).It is easy to see that a G δ -set is an S δ -set. Hence the notion of S δ -set is a generalization of G δ -set. See [3] for a description of S-normal spaces whose closed subsets are S δ -sets.