Hausdorff characterizations of first countable T1 spaces via fixed point theorems

TL;DR

This paper characterizes first countable T1 spaces as Hausdorff spaces using fixed point theorems for set-valued maps with contractive orbits, and applies these results to optimization and intersection theorems.

Contribution

It introduces new notions of contractive orbits in first countable spaces and links them to Hausdorff properties via fixed point theorems, extending classical results.

Findings

Hausdorff property characterized via fixed point theorems

Derived a sufficient condition for a function to attain a strong minimum

Generalized Cantor's intersection theorem for nested sets with shrinking diameters

Abstract

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.