Urysohn separation property in topological molecular lattices

A new approach to -, -Urysohn, and -completely Hausdorff axioms

“…Urysohn axiom had been generalized to L-topological spaces and topological molecular lattices by Chen and Wu [1] and Fang and Yue [6], respectively. But in general the L-real line and the L-unit interval need not satisfy this axiom since they need not satisfy T 2 axiom in [22,42] (see Example 5.10).…”

“…Therefore [0, 1](I ) does not satisfy Hausdorff axiom. Further we know that E(I ) and R(I ) do not satisfy U 2 axiom in the sense of [1], Urysohn axiom, and H ( )-complete Hausdorff axioms in the sense of [5,6].…”

“…There have been diverse studies on separation axioms in L-topology (see [1,[5][6][7][11][12][13]16,[18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][40][41][42][43][44][45] etc.). Some of them were only motivated by extending the syntax of separation axioms in general topology; while others were motivated by the relations between separation axioms and compactness, uniformities, metrics, convergence, etc.…”

“…Chen and Wu generalized the Urysohn axiom to topological molecular lattices and called it the U 2 axiom [1]. Fang also generalized them to L-topological spaces and topological molecular lattices in [5,6].…”

A note on “Urysohn separation property in topological molecular lattices”

“…In a topological molecular lattice (L(M), ), let e ∈ M and P ∈ , then P is called a remote-neighborhood of e if e P . The set of all remote-neighborhoods of e is denoted by (e).In order to extend Urysohn separation axioms to topological molecular lattices, Chen and Wu introduced the following definition:Definition 1 (Chen and Wu [2]). Let (L(M), ) be a TML.…”