We introduce the notion of a strictly completely regular bitopological space and show that the category of strictly completely regular bitopological spaces is isomorphic to the category of strictly completely regular ordered spaces. It is shown by means of examples that the category of strictly completely regular bitopological spaces is more flexible than its counterpart in ordered spaces. We also determine quasiuniformities that induce strictly completely regular bitopologies.

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In this paper we show that the bitopological analogue of the Stone-Čech compactification can be obtained from the Nachbin ordered compactification when restricted to strictly completely regular bitopological spaces.

We also study conditions for complete quasi-pseudometrizability.2000 Mathematics Subject Classification. 54F05, 54E35, 54C35. Introduction. An ordered topological space (or simply, an ordered space) is a triple (X, τ, ≤), where τ is a topology on X and ≤ is a closed partial order on X. We will use R 0 to denote the set of real numbers with the usual topology and the usual order. For a subset A of the set of real numbers, we use A 0 to denote A as an ordered subspace of R 0 , for example I 0 denote the closed interval with the usual topology and the usual order.

We discuss the monad associated with the topology of pointwise convergence. We also study examples of the Eilenberg-Moore algebras for this monad.