This paper begins the study of monotone normality, a common property of linearly ordered spaces and of Borges' stratifiable spaces. The concept of monotone normality is used to give necessary and sufficient conditions for stratifiability of a T.-space, to give a new metrization theorem for p-spaces with G g-diagonals, and to provide an easy proof of a metrization theorem due to Treybig. The paper concludes with a list of examples which relate monotone normality to certain familiar topological properties. 1. Introduction. The property of monotone normality first appears, without name, in Lemma 2.1 of C. R. Borges' paper Orz stratifiable spaces [5]. In [29], P. Zenor named the property and announced results relating monotone normality to metrizability and stratifiability. Subsequently Heath and Lutzer ([17], [18]) and Borges [7] announced results complementary to Zenor's original work, showing, in particular, that monotone normality unexpectedly holds in one large class of spaces-the linearly ordered spaces-but fails to hold in others where it might be expected. This paper is a combination of the independent studies conducted by Zenor and by Heath and Lutzer. § §2 through 4 are the work of Zenor, with the exception of Lemma 2.2 which, together with § §5 through 7, is due to Heath and Lutzer. The authors would like to thank the editor for suggesting this format. 2. Definitions and preliminary results. Throughout this paper all spaces are assumed to be at least T and "mapping" means "continuous onto function." The set of natural numbers is denoted by the letter zV. 2.1. Definition. A space X is monotonically normal if there is a function G which assigns to each ordered pair ÍH, K) of disjoint closed subsets of X an open set G(/7, K) such that (a) H C GÍH, K) C G(W, K)~ C x\K; (b) if (H , K) is a pair of disjoint closed sets having H C H and K D K

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MSC: primary 54D20 secondary 54E30, 54E35, 54F05 Keywords: Metacompact Countably metacompact Monotonically metacompact Monotonically countably metacompact Generalized ordered space GO-space LOTS Metacompact Moore space Metrizable σ -Closed-discrete dense setWe show that any metacompact Moore space is monotonically metacompact and use that result to characterize monotone metacompactness in certain generalized ordered (GO) spaces. We show, for example, that a generalized ordered space with a σ -closeddiscrete dense subset is metrizable if and only if it is monotonically (countably) metacompact, that a monotonically (countably) metacompact GO-space is hereditarily paracompact, and that a locally countably compact GO-space is metrizable if and only if it is monotonically (countably) metacompact. We give an example of a non-metrizable LOTS that is monotonically metacompact, thereby answering a question posed by S.G. Popvassilev. We also give consistent examples showing that if there is a Souslin line, then there is one Souslin line that is monotonically countable metacompact, and another Souslin line that is not monotonically countably metacompact.