Topological kernel of sets and application on fractals

Document Type : Original Article

Authors

1 Department of Mathematics, Faculty of Science, Tanta University, Tanta, Egypt.

2 Department of Physics and Engineering Mathematics, Faculty of Engineering, Kafrelsheikh University, Egypt.

Abstract

In various mathematical sciences, sets and functions in topology have been extensively developed and exploited. Some novel separation axioms have been discovered through studying generalizations of closures duo to closed sets, $\bigwedge$-sets, and $\bigvee$-sets. Self similar fractals have important role in some real life problems as physics and engineering. In this paper, the topology described by the family of $\delta\gamma$-$\bigwedge$ and $\delta\gamma$-$\bigvee$ sets in topological spaces is defined and studied in terms of $\bigwedge_{\delta\gamma}$-Sets and $\bigvee_{\delta\gamma}$-Sets. Also, the topological space $\mathbf{Top_{(\X,\tau^{\bigwedge_{\delta\gamma}})}}$ is defined and studied. Additionally, several features of these sets are presented, as well as some associated new separation axioms. Finally, we improved theorems 4.3 \cite{CaldasJavNoi2000} and 6.5 \cite{Caldas2005} for Caldas et al. We approximate self similar fractals through graph theory to topological spaces. Some topological properties such as separation axioms are studied. Finally, the kernel of topological approximations of fractals are calculated in terms of their connecting points.

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