Document Type : Research Paper

Authors

• Somaye Grailoo ¹
• Abasalt Bodaghi ²
• Abolfazl Niazi Motlagh ³

¹ Esfarayen University of Technology, Esfarayen, North Khorasan, Iran

² Department of Mathematics, West Tehran Branch, Islamic Azad University, Tehran , Iran

³ Department of Mathematics, Faculty of Basic Science, Univercity of Bojnord, P. O. Box 1339, Bojnord, Iran

Abstract

A metric modular on a set $X$ is a function $w : (0,\infty)\times X\times X\longrightarrow  [0,\infty]$ written as $(\lambda,x,y)\mapsto w_{\lambda}(x,y)$ satisfying, for all $x, y, z\in X$, the following three properties: $x = y$  if and only if  $w_{\lambda}(x, y) = 0$ for all $\lambda>0$; $w_{\lambda}(x, y) = w_{\lambda}(y, x)$ for all $\lambda>0$; $w_{\lambda+\mu}(x, y) \leq w_{\lambda}(x, z) + w_{\mu}(y, z)$ for all $\lambda, \mu>0$. In this paper we define a Hausdorff topology on  metric modular spaces and prove  some known results of metric spaces including  Baire's theorem and  Uniform limit theorem for metric modular spaces.

Keywords

• modular
• metric modular
• Baire’s theorem
• Uniform limit theorem