Advances in Pure Mathematics, 2015, 5, 552-559 Published Online July 2015 in SciRes. http://www.scirp.org/journal/apm http://dx.doi.org/10.4236/apm.2015.59051 How to cite this paper: El-Saady, K. and Al-Nabbat, F. (2015) Generalized Topological Molecular Lattices. Advances in Pure Mathematics, 5, 552-559. http://dx.doi.org/10.4236/apm.2015.59051 Generalized Topological Molecular Lattices Kamal El-Saady 1 , Fatima Al-Nabbat 2 1 Department of Mathematics, Faculty of Science at Qena, South Valley University, Qena, Egypt 2 Department of Mathematics, College of Science, King Faisal University, Al-Hasa, Saudi Arabia Email: el-saady@lycos.com , fatima_math20@yahoo.com Received 11 May 2015; accepted 7 July 2015; published 10 July 2015 Copyright © 2015 by authors and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/ Abstract The aim of this paper is to introduce the concept of generalized topological molecular lattices briefly GTMLs as a generalization of Wang’s topological molecular lattices TMLs, Császár’s set- point generalized topological spaces and lattice valued generalized topological spaces. Some no- tions such as continuous GOHs, convergence theory and separation axioms are introduced. More- over, the relations among them are investigated. Keywords Generalized Topological Molecular Lattices, Generalized Order Homomorphisms, Convergence of Molecular Nets, Separation Axioms 1. Introduction In 1992 Wang [1], introduced his important theory called topological molecular lattice (briefly, TML) as a generalization of ordinary topological and fuzzy topological spaces in tools of molecules, remote neighborhoods and generalized order homomorphisms GOHs. Then many authors characterized some topological notions in such TMLs, such as convergence theories of molecular nets or ideals [1]-[3], separation axioms [1] [4] and other notions. In this paper, we aim to introduce a generalization of TMLs under the name of generalized topological molecular lattice (briefly, GTML). In the same manner, we study several notions in these GTMLs, investigate some properties and set the relations among these notions including GOHs, convergence theories and separation axioms. Throughout this work, ( ) , , , L ′ V is a complete lattice with an order-reversing involution () ' , and with the smallest element ⊥ and the largest element  ( ) ≠⊥  . By an L-generalized topology [5], on a non-empty ordinary set X, we mean a subfamily τ of X L with the following axioms: