The Urysohn Spectrum: A Metric for Compactness and Connectedness

This paper introduces the concept of the Urysohn Spectrum as a novel metric for quantifying topological properties, specifically compactness and connectedness. Traditional topological definitions characterize these properties qualitatively, often as binary attributes. By leveraging Urysohn's Metrization Theorem, which enables the embedding of a wide class of topological spaces into metric spaces, we propose a method to associate a measurable "spectrum" of distances with any given topological space. This Urysohn Spectrum is defined as the set of all distances between points within the embedded image of the space in a universal metric space, such as the Hilbert cube. We hypothesize that the structural properties of this spectrum, particularly its boundedness and its characteristic of being an interval in the real line, serve as quantitative indicators of the original space's compactness and connectedness, respectively. The methodology outlines the construction of such a spectrum through Urysohn functions and the induced metric. Theoretical results suggest that a compact space yields a bounded and closed Urysohn Spectrum, while a connected space yields a Urysohn Spectrum that is an an interval. This framework offers a pathway to bridge qualitative topological concepts with quantitative metric analysis, providing new tools for studying and classifying complex spaces.