Abstract: Replacing the set of real numbers by an ordered Banach space in the definition of a metric, Guang and Xian [5] introduced the concept of a cone metric and obtained some fixed point Theorems for contractive mappings on cone metric spaces. It has been shown that every cone metric space is metrizable [2-4]. In this paper we review and simplify some results of [6] and as a consequence of ...

If 〈L, <〉 is a dense linear ordering without end points and A and B disjoint dense subsets of L, then the topology OAB on the set L generated by closed intervals [a, b], where a ∈ A and b ∈ B, is finer than the standard topology, O<, generated by all open intervals and 〈L,OAB〉 is a GO-space. The basic properties of the topology OAB (separation axioms, cardinal functions, metrizability) are inve...

In this paper we discuss the question ”When do different orderings of the rational function field R(X) (where R is a real closed field) induce the same R-place?”. We use this to show that if R contains a dense real closed subfield R′, then the spaces of R-places of R(X) and R′(X) are homeomorphic. For the function field K = R(X) we prove that its space M(K) of R-places is metrizible if and only...

In this article, we will be interested in the extent to which the assumption of first countability in this theorem can be weakened. Recall that a Hausdorff topological space X is Fréchet if whenever x is a limit point of A ⊆ X, there is a sequence an (n < ω) of elements of A which converges to x. This is a natural weakening of first countability which has been extensively studied in the literat...

Let Y be an absolute neighbourhood retract (ANR) for the class of metric spaces and let X be a Hausdorff space. Let Y X denote the space of continuous maps from X to Y equipped with the compact open topology. It is shown that if X is a CW complex then Y X is an ANR for the class of metric spaces if and only if Y X is metrizable and has the homotopy type of a CW complex. The same holds also when...