Abstract. This paper considers the problem of defining Cauchy sequence and completeness in quasi-pseudo-metric spaces. The definitions proposed allow versions of such classical theorems as the Baire Category Theorem, the Contraction Principle and Cantor's characterization of completeness to be formulated in the quasi-pseudo-metric setting.
In this note the converses of recent fixed-point theorems due to KANNA~ and CHA~ERJEA are obtained. An example is constructed to show that a metric space having the fixed-point property for homeomorphisms need not be metrically topologically complete. An example of COW,CELL is formulated in a more general perspective. w 1. Introduction II~r [5] showed that a metric space is complete if and only if any contraction on closed subsets thereof has a fixed-point. In this context, it is easily seen from an example due to CON~ELL ([3], p. 978, Example 3) that it is not however possible to conclude that a metric space is complete if any contraction on it has a fixed-point. In fact, the fixed-point property for even continuous maps does not insure the completeness of the metric space. Besides ttv 's [5], there are results converse to the contraction mapping principle. But mostly these assert the existence of complete metric topologies such that a class of mappings of an abstract set into itself with fixed-points consists of contractions (see e. g. [1]). Theorems t and 2 of Section 2 of this note on the other hand have in their conclusions the completeness of the metric space under a hypothesis that each member of a class of mappings with constraints on transformation of distance has fixed or periodic points. Incidentally it subsumes converses of recent results ([2], [6]) in the form of fixed-point theorems. Theorem 1 is independent of Hc's result, mentioned above.The author thanks the referee for drawing his attention to the note of Hv [5].The connection between metric-topological completeness of a metric space (i. e. the existence of a metric whose topology is the
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