On $\theta$-Modifications of generalized topologies via hereditary classes
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Let $(X, \mu)$ be a generalized topological space (GTS) and $\h$ be a hereditary class on $X$ due to Cs\'asz\'ar \cite{2csas2008}. In this paper, we define an operator $()^{\circ}: \pp(X)\rightarrow \pp(X)$. By setting $c^{\circ}(A)=A\cup A^{\circ}$ for every subset $A$ of $X$, we define the family $\mu^{\circ}=\{M\subseteq X: X-M=c^{\circ}(X-M)\}$ and show that $\mu^{\circ}$ is a GT on $X$ such that $\mu(\theta) \subseteq\mu^{\circ}\subseteq \mu^{*}$, where $\mu^{*}$ is a GT in \cite{2csas2008}. Moreover, we define and investigate $\mu^{\circ} $-codense and strongly $\mu^{\circ} $-codense hereditary classes.
Abstract
Let $(X, \mu)$ be a generalized topological space (GTS) and $\h$ be a hereditary class on $X$ due to Cs\'asz\'ar \cite{2csas2008}. In this paper, we define an operator $()^{\circ}: \pp(X)\rightarrow \pp(X)$. By setting $c^{\circ}(A)=A\cup A^{\circ}$ for every subset $A$ of $X$, we define the family $\mu^{\circ}=\{M\subseteq X: X-M=c^{\circ}(X-M)\}$ and show that $\mu^{\circ}$ is a GT on $X$ such that $\mu(\theta) \subseteq\mu^{\circ}\subseteq \mu^{*}$, where $\mu^{*}$ is a GT in \cite{2csas2008}. Moreover, we define and investigate $\mu^{\circ} $-codense and strongly $\mu^{\circ} $-codense hereditary classes.
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