商位相順序空間について

In the theory of general topology, the following theorem is well known(c. f. [2] or [4]) For a topological space X, and an equivalence relation R on X, if the quotient space X/R is Hausdorff, then R is closed in the product space X^2 . If the projection p of a space X onto the quotient space X/R is open, and R is closed in X^2, then X/R is a Hausdorff space. The analogy of this theorem in a topological ordered space has been obtained in the case where X is a compact ordered space (c. f. [9] Proposition 9). In this paper, we shall study the sufficient conditions for X/R to be T_2-ordered, and give some examples. For the problem of this kind, S. D. McCartan studied in [6] a particular quotient ordered space (that is, a quotient ordered space by a particular equivalence relation) which inherites some interesting properties of the domain ordered space.
The author wishes to express his gratitude to Professor Osamu Takenouchi for his helpful suggestions and encouragement in the preparation of this paper.