Abstract
In [18], Tallafha and Khalil introduced a new type of uniform space which is a mixture of topology and analysis, namely semi-linear uniform spaces. In this paper, we continue and extend this study by presenting a new generalization of a semi-linear uniform space, namely weak semi-linear uniform spaces. Then we present the concept of proximinal sets without any tools that the metric structure usually offers. We show that every finite set and every convergent sequence with its limit are proximinal sets. Furthermore, we give a characterization of metrizable spaces and we study some of the topological properties of weak semi-linear uniform spaces.
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