A method of returning vector-valued maps to real-valued functions on monotone operators

Let Y be an ordered topological vector space with a positive interior point e. Motivated by the Minkowski functional pe, we introduce the function qe on the interior intYY+ of the positive cone of Y. The function qe plays a role for evaluating a 'reverse gauge' of the Minkowski functional pe on the restricted space intYY+. We also apply the function qe for giving complete proof of the recent results [7] by Jin, Xie and Yue on monotonically countably paracompact spaces. New characterizations by vector-valued continuous maps are also given for stratifiable spaces and monotone cb-spaces, the latter provides a positive answer to a question posed in [12] by the author.