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On the topology and $$wt$$-distance on metric type spaces. (English) Zbl 1333.54035
Summary: Recently, M. A. Khamsi and the first author [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 73, No. 9, 3123–3129 (2010; Zbl 1321.54085)] discussed a natural topology defined on any metric type space and noted that this topology enjoys most of the metric topology like properties. In this paper, we define topologically complete type metrizable space and prove that being of metrizability type is preserved under a countable Cartesian product and establish the fact that any set in a complete metric type space is a topologically metrizable type space. Next, we introduce the concept of $$wt$$-distance on a metric type space and prove some fixed point theorems in a partially ordered metric type space with some weak contractions induced by the $$wt$$-distance.

##### MSC:
 54E35 Metric spaces, metrizability 54H25 Fixed-point and coincidence theorems (topological aspects) 54F05 Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces
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