Journey into special $T_1$-spaces

In this survey, we review certain types of special $T_1$ spaces and their associated fixed-point theorems. Kupka introduced the notion of a feeble topological contraction, which naturally generalizes Lipschitz contractions defined on metric spaces. Specifically, Kupka established a fixed-point theorem for feeble topological contractions possessing a closed graph within the product of arbitrary $T_0$ spaces. Furthermore, the $T_1$ separation axiom is shown to guaranty the uniqueness of such fixed points. Finally, we discuss peripheral Hausdorff and locally Hausdorff spaces within the context of these fixed-point results.