On P-Contractions in Hausdorff Spaces with a τ-Distance

In this paper, we obtain fixed point results in Hausdorff topological spaces using the notion of τ-distance. We introduce a new type of contraction, called a P-contraction, defined on topological spaces equipped with a τ-distance, and establish a fixed point theorem for self-maps satisfying this condition. Assuming that the space is p-bounded and S-complete, and that the mapping is p-τ-continuous, we prove the existence and uniqueness of fixed points for such self-maps. Our results generalize several well-known fixed point theorems in the literature by both weakening the classical assumptions via the topological structure and by extending the contraction condition. In addition to an example that supports this generalization, we present another that highlights the advantages of employing the τ-distance function instead of the standard metric. Finally, we briefly discuss how certain metric fixed point theorems from the literature can be derived using the τ-distance framework.