The author introduces the concept of

$F$-type topological space and gives a characterization of this kind of spaces. He points out that the usual metric spaces, Hausdorff topological vector space and Menger space are all special cases of

$F$-type space. Using this concept, some fixed point theorems and a variational principle in

$F$-type topological space are established. As an application, the author utilizes the results presented in this paper to obtain a variational principle and a fixed point theorem in Menger probabilistic metric spaces, which generalize the corresponding results of

*I. Ekeland* [ibid. 47, 324-353 (1974;

Zbl 0286.49015)],

*J. Caristi* [Trans. Am. Math. Soc. 215, 241-251 (1976;

Zbl 0305.47029)] and the reviewer with

*Chen Yuqing* and

*Guo Jinli* [Acta Math. Appl. Sin., Engl. Ser. 7, No. 3, 217-228 (1991;

Zbl 0743.54017)].