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.

💡 Research Summary

The paper investigates the relationship between the Hausdorff property of first‑countable (T_{1}) spaces and fixed‑point theorems for set‑valued maps whose orbits exhibit a form of contraction. Two distinct notions of a “contractive orbit” are introduced. The first, called a (\tau)-contractive orbit, is defined purely in topological terms: for any open cover of the space there exists a member of the cover that contains some point of the orbit together with the entire image of that point under the set‑valued map. The second, a (p)-contractive orbit, is defined with respect to a generalized distance function (p) (a premetric) that need not be symmetric nor satisfy the triangle inequality. Both notions are accompanied by an auxiliary condition denoted (\bar{\star}), which requires that whenever a subsequence of the orbit converges to a limit (x) and a point (y\in S(x)) with (y\neq x) exists, then one can extract a further subsequence whose images converge to (y). This condition captures the idea that the orbit continues to “shrink” towards any point that lies in the image of its limit.

The central results are two theorems establishing an equivalence between Hausdorffness and a fixed‑point property for such orbits. Theorem 2.6 shows that if the underlying space ((X,\tau)) is Hausdorff, any infinite (\tau)-contractive orbit satisfying (\bar{\star}) possesses a strong accumulation point (\bar{x}) (i.e., a point such that both the orbit points and their images eventually lie in every neighbourhood of (\bar{x})). Moreover, the theorem proves that (S(\bar{x})\subseteq{\bar{x}}); in other words, (\bar{x}) is a fixed point of the set‑valued map. The proof uses Lemma 2.4 to guarantee the existence of a strong accumulation point, then exploits the Hausdorff separation to derive a contradiction if (S(\bar{x})) contained a distinct point, thereby forcing the inclusion.

Conversely, Theorem 2.7 asserts that if for every set‑valued map (S) and every (\tau)-contractive orbit satisfying (\bar{\star}) the corresponding strong accumulation point is a fixed point, then the space must be Hausdorff. The argument proceeds by assuming non‑Hausdorffness, constructing (via Lemma 2.5) a sequence that converges simultaneously to two distinct points (\bar{x}) and (\bar{y}), and defining a deliberately pathological map (S) that swaps these two limit points while preserving the (\tau)-contractive and (\bar{\star}) properties. This yields a contradiction, establishing that the fixed‑point condition forces Hausdorff separation.

These two theorems together provide a new characterization of Hausdorffness: a first‑countable (T_{1}) space is Hausdorff if and only if every (\tau)-contractive orbit with the (\bar{\star}) property has a fixed point. This characterization does not rely on completeness or metric structure, thereby extending classical fixed‑point results (such as Banach’s contraction principle) to a purely topological setting.

The paper then derives several notable applications. Corollary 2.10 uses the main theorem to give a sufficient condition for a proper, lower‑semicontinuous-from‑above function (f) to attain a strong minimum: if the associated descent map (S_f(x)={y\mid f(y)<f(x)}) admits a (\tau)-contractive orbit, then the limit point of that orbit is a minimizer of (f). The proof verifies that the decreasing sequence of function values ensures the (\bar{\star}) condition, and then applies Theorem 2.6 to conclude (S_f(\bar{x})=\emptyset), i.e., (\bar{x}) is a minimum.

Corollary 2.11 generalizes Cantor’s intersection theorem. For a nested family of non‑empty closed sets ({C_i}) whose diameters tend to zero, the authors construct a set‑valued map that forces a (\tau)-contractive orbit to visit deeper and deeper members of the family. Theorem 2.6 then guarantees a unique point belonging to all (C_i), reproducing the classical intersection result without assuming metric completeness.

Section 3 introduces premetric spaces (or (p)-spaces), where a function (p:X\times X\to\mathbb{R}+) satisfies three axioms: (i) (p(x,y)=0) iff (x=y); (ii) convergence of (p(x,x_n)) to zero implies (x_n\to x); (iii) ordinary convergence of a sequence forces successive distances (p(x{n+1},x_n)) to vanish. Within this framework, a (p)-contractive orbit is defined analogously to the LOEV principle. The authors prove that in a Hausdorff (p)-space, any (\tau)-contractive orbit is automatically (p)-contractive, though the converse fails in general. This comparison clarifies the relative strength of the two contraction notions and situates the results within the broader literature on generalized metric spaces.

Section 4 returns to the metric case, showing that when (p) is a genuine metric the two notions of contractive orbit coincide. Moreover, the authors answer an open question from