Beyond Open Sets: Topological Spaces Via Near and Far When most people think of topology, they picture rubber sheets, coffee mugs turning into donuts, or the quirky world of Möbius strips. While these are fun entry points, modern applied topology is undergoing a quiet revolution. Instead of just asking “Can we deform A into B?” , a more pragmatic question is emerging: What is near and what is far in a topological space? Welcome to the perspective of Topology with Applications: Topological Spaces via Near and Far . The Classical Foundation: Open Sets Traditionally, a topological space is defined by a collection of open sets. Two points are "near" if they share an open neighborhood. This is precise, but it's also qualitative. It tells us that closeness exists, but not how close . For applications in data science, digital imaging, or sensor networks, we need a sharper lens. We need to answer:
Is this data point near that cluster? Is this pixel far from an edge boundary? How do we measure the gap between two disjoint sets?
Redefining Proximity: The Nearness Relation Enter the concept of near sets . Two subsets ( A ) and ( B ) of a topological space are considered near if their closures intersect: [ A \ \delta \ B \ \iff \ \text{cl}(A) \cap \text{cl}(B) \neq \emptyset ] But this is just the start. In applied near set theory (pioneered by Peters, Naimpally, et al.), we refine this using descriptions . Instead of just points, we consider features. A probe function measures a property of a point (e.g., color, temperature, intensity). Two sets are near if they have points with matching descriptions.
Example in Image Processing: Let ( A ) be a group of pixels in a retinal scan, and ( B ) a group in a stored template. They are near if their feature vectors (intensity + texture + edge orientation) are within a threshold ( \varepsilon ). Beyond Open Sets: Topological Spaces Via Near and
This shifts topology from pure geometry to perceptual tolerance . The Dual: Farness (Remoteness) What is far? In classical topology, disjoint closed sets can still be "near" in the sense of having no open separation. But in applications, far means distinguishable or remote. We say ( A ) is far from ( B ) if there exists an ( \varepsilon > 0 ) such that for every ( a \in A ) and ( b \in B ), the descriptive distance exceeds ( \varepsilon ). In practical terms:
Two clusters in a scatter plot are far if no reasonable radius connects them. Two textures in an image are far if no probe function yields a match.
Farness allows us to define boundaries , detect novelty , and separate clusters . Why This Matters: Real-World Applications This "near and far" lens transforms topology into a computational tool: Welcome to the perspective of Topology with Applications:
Digital Image Analysis Near sets identify regions of interest (e.g., tumors) by finding pixels near a target description. Far sets define backgrounds or irrelevant artifacts.
Sensor Networks Each sensor reports a description (temperature, vibration, humidity). Nodes are near if their descriptions match within tolerance — forming dynamic clusters without predefining a metric.
Machine Learning Topological data analysis (TDA) often uses persistent homology. Adding near/far relations helps explain why two classes are separable or why a point is an outlier. This is precise, but it's also qualitative
Medical Diagnosis Compare a patient’s tissue sample (set of cell descriptions) to a database of healthy vs. diseased samples. Nearness to the diseased group triggers a flag.
A Simple Example Consider the unit interval ([0,1]) with the usual topology. Let: