In 2001 the authors had introduced different L-separation axioms denoted by GTi, i =
0; 1; 2; 3; 4. In this and a subsequent paper, we introduce a notion of completely regular L-
topological spaces depend on the notion of L-numbers presented by S. G¨ahler and W. G¨ahler
in 1994. We denote by GT31
2
-space (or L-Tychonoff space) the L-topological space which is GT1
and completely regular in this sense. The category L-Tych of GT31
2
-spaces is topological over the
category Set of sets, that is, the initial and final lifts and also the initial and final GT31
2
-spaces
exist in L-Tych. Moreover, the relation between the GT31
2
-spaces, GT4-spaces and GT3-spaces,
goes well. It is also shown here that our completely regular spaces are more general than the completely
regular spaces defined by Hutton in 1975, Katsaras in 1980, and by Kandil and El-Shafee
in 1988. The relation of the GT31
2
-spaces with the L-proximity spaces, the L-uniform spaces and
the L-compact spaces will be investigated in part II of this paper. Moreover, the relation between
the GT31
2
-spaces and the L-topological groups will be studied in a separate paper.
Keywords: Fuzzy filters; GTi-spaces; completely regular spaces; GT31
2
-spaces; L-Tychonoff spaces;
Initial and final lifts; Initial and final L-topological spaces; L-proximity spaces; L-uniform spaces;
L-compact spaces; L-topological groups.