利用Chittenden度量化定理证明$b$ -度量空间和$θ$ -度量空间的度量化
Metrizability of $b$-metric space and $θ$-metric space via Chittenden's metrization theorem
摘要：
在[An，V.T.，Tuyen，Q.L.，Dung，V.N.，Stone-type theorem on $b$ - metric spaces and applications，Topology Appl.185 -186（2015）50-64]，Tran Van An等人提供了$b$ -度量空间可度量化的充分条件。他们通过假设距离函数在一个变量中连续来证明可度量化。本文的主要目的是对Khamsi和Hussain在$b$ [1，Khamsi，M. A和Hussain，N.，度量型空间中的KKM映射，非线性分析。73（9）（2010）3123-3129]{kh}通过Chittenden的度量化定理，而无需对距离函数的任何假设。本文进一步证明了Khojasteh等人在[Khojasteh，F.，Karapinar，E.，Radenovic，S.，$θ$ -度量空间：一个推广，工程中的数学问题，第2013卷，文章504609，7页]。
英文摘要：
In [An, V.T., Tuyen, Q.L., Dung, V.N., Stone-type theorem on $b$-metric spaces and applications, Topology Appl. 185-186 (2015) 50-64], Tran Van An et al. provide a sufficient condition for $b$-metric space to be metrizable. They proved the metrizability by assuming that the distance function is continuous in one variable. The main purpose of this manuscript is to provide a direct short proof of the metrizability of $b$-metric space introduced by Khamsi and Hussain in \cite[\, Khamsi, M.A and Hussain, N., KKM mappings in metric type spaces, Nonlinear Anal. 73 (9) (2010) 3123-3129]{kh} via Chittenden's metrization theorem without any assumption on the distance function. Further in this short note, we prove the metrizability of $θ$-metric space introduced by Khojasteh et al. in [Khojasteh, F., Karapinar, E., Radenovic, S., $θ$-metric space: A Generalization, Mathematical problems in Engineering, Volume 2013, Article 504609, 7 pages].
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