Strongly Generalized Neighborhood Systems

Abstract

In this paper we define strongly generalized neighborhood systems (in brief strongly $GNS$) and study their properties. It's proved that every generalized topology $\mu$ on $X$ gives a unique strongly $GNS$ $\psi_{\mu}:X\rightarrow \exp{(\exp{X})}$. We prove that if a generalized topology $\mu$ is given, then $\mu_{\psi_{\mu}}=\mu$; and if a strongly $GNS $ $\psi$ is given, then $\psi_{\mu_{\psi}}=\psi$. Strongly $(\psi_{1},\psi_{2})$-continuity is defined. We prove that $f:X\rightarrow Y$ is strongly $(\psi_{1},\psi_{2})$-continuous if and only if it is $(\mu_{\psi_{1}},\mu_{\psi_{2}})$-continuous.