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A space is defined to be suborderable if it is embeddable in a (totally) orderable space. It is shown that a metrizable space X is suborderable iff (1) each component of X is orderable, (2) the set of cut points of each component of X is open, and (3) each closed subset of X which is a union of components has a base of clopen neighborhoods. Note that condition (1) and hence this result is topological since there are many good topological characterizations of connected orderable spaces. In a space X let Q denote the union of all nondegenerate components each of whose noncut points has no compact neighborhood. It is also shown that a metrizable space X is orderable iff (1) X is suborderable, (2) X − Q X - Q is not a proper compact open subset of X, and (3) if W is a neighborhood of p ∈ X p \in X and K is the component in X containing p such that ( W − K ) − Q (W - K) - Q has compact closure and { p } \{ p\} is the intersection of the closures of ( W − K ) − Q (W - K) - Q and ( W − K ) ∩ Q (W - K) \cap Q , then K is a singleton. Corollaries are given; every condition in each of these corollaries is concisely stated and sufficient for a space to be orderable when it is metrizable and suborderable. Both of these results are extended to a class properly containing the metrizable spaces.

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Abstract.The Stone-Cech compactification of a Tychonoff space is orderable iff it is a pseudocompact suborderable space.Recently, M. Venkataraman, M. Rajagopalan, and T. Soundararajan showed [V-R-S] that a necessary condition for the Stone-Cech compactification of a Tychonoff space to be orderable is that it be normal and countably compact. They also gave several sufficient conditions.The purpose of this paper is to show that in the above theorem the condition is both necessary and sufficient if normal is changed to suborderable. This result was announced in [P2]. Also see [PI].Recently, this result was obtained independently by J. H. Weston.The following proof is a shorter version of that which appeared in [PI] using a result of Gillman and Henriksen [G-H]. Definition 1. A space X is suborderable if there is a total orderinĝ on its elements such that : (1) The original topology on X is finer than its open interval topology. (2) X has a basis consisting of (possibly degenerate) intervals. Under these conditions ^ is called an admissible order on X. Definition 2. A subordered space is a pair (A', ^) where X is a suborderable space and ^ is an admissible order on X.Cech, who introduced the class of suborderable spaces, proved that a space is suborderable iff it can be embedded in an (totally) orderable space.The following definitions are equivalent to those that appear in Definition 9.3 of [G-H].Definition 3. Let X be an ordered space, and let cua be an infinite regular initial ordinal. A monotone sequence 5'={xç}ç<(l) of points in X is called a Q sequence if {^}ç

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