Hausdorff characterizations of first countable T1 spaces via fixed point theorems

We introduce two notions of a contractive orbit of a set-valued map defined in a first countable space. The first defines the contraction with respect to the topology of the underlying space while the second defines the contraction with respect to a generalized distance function. We characterize the Hausdorff property of first countable $T_1$ spaces via fixed point theorems for set-valued maps with a contractive orbit satisfying some additional assumptions. As an application, we derive a sufficient condition for a function to attain a strong minimum and generalize Cantor’s intersection theorem for a sequence of closed nested sets with diameters converging to 0.