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Hausdorff characterizations of first countable T1 spaces via fixed point theorems

Detelina Kamburova

TL;DR

This work characterizes when a first countable $T_1$ space is Hausdorff by linking fixed-point principles for set-valued maps with contractive orbits to the underlying topology. It introduces two contractive notions—a topology-based $ au$-contractive orbit and a generalized-distance-based $p$-contractive orbit—and develops LOEV-inspired arguments to show that, under appropriate orbit properties, accumulation points must be fixed in a precise sense. The paper first treats general topological spaces, then premetric (p-space) settings, and finally the metric case, establishing equivalences and implications across these frameworks. Notable contributions include sufficient conditions for strong minima, Cantor-type intersection generalizations in non-metric contexts, and an answer to Kupka's question by showing when a topological contraction becomes a Banach contraction in the metric setting.

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.

Hausdorff characterizations of first countable T1 spaces via fixed point theorems

TL;DR

This work characterizes when a first countable space is Hausdorff by linking fixed-point principles for set-valued maps with contractive orbits to the underlying topology. It introduces two contractive notions—a topology-based -contractive orbit and a generalized-distance-based -contractive orbit—and develops LOEV-inspired arguments to show that, under appropriate orbit properties, accumulation points must be fixed in a precise sense. The paper first treats general topological spaces, then premetric (p-space) settings, and finally the metric case, establishing equivalences and implications across these frameworks. Notable contributions include sufficient conditions for strong minima, Cantor-type intersection generalizations in non-metric contexts, and an answer to Kupka's question by showing when a topological contraction becomes a Banach contraction in the metric setting.

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 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.
Paper Structure (4 sections, 17 theorems, 14 equations)

This paper contains 4 sections, 17 theorems, 14 equations.

Table of Contents

  1. Introduction
  2. Hausdorff characterization in general topological spaces
  3. Characterization of Hausdorffness in premetric spaces
  4. Metric case

Key Result

Lemma 2.3

Let $(X, \tau)$ be a first countable Hausdorff space. Let $S: X \rightrightarrows X$ be a set-valued map and the $S$-orbit $M:=\{x_i\}_{i=1}^{\infty}$ be $\tau$-contractive. Then there exists an accumulation point of $M$, $\bar{x} \in X$, such that: any neighbourhood of $U_{\bar{x}}$ of $\bar{x}$ is and whenever $a_k \in S(x_{i_k})$ as well.

Theorems & Definitions (29)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Example 2.7
  • Corollary 2.8
  • Corollary 2.9
  • Theorem 2.10
  • ...and 19 more