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 $w:(0,\mathrm{\infty })\times X\times X⟶[0,\mathrm{\infty }]$ 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 $\lambda >0$; $w_{\lambda +\mu }(x,y)\le w_{\lambda }(x,z)+w_{\mu }(y,z)$ 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