Topology With Applications Topological Spaces Via Near And Far -

Consider a field of sensors measuring temperature or motion. A sensor (s_i) is near (s_j) if they can communicate (signal strength > threshold). But more useful: a set of sensors (A) is near an event region (B) if the collective readings can detect it.

(2016): Explores the classification of levels of nearness between sets in topological spaces. About the Book The book itself replaces traditional

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. Consider a field of sensors measuring temperature or motion

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.

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. (2016): Explores the classification of levels of nearness

: Digital image processing, pattern recognition, and forgery detection. Biology : Population dynamics and stem cell biology. Engineering : Camouflage filters and visual merchandising. Topological Spaces Via Near And Far [PDF] - VDOC.PUB

The human visual system groups elements based on Gestalt principles: proximity, similarity, continuity. These are proximities in feature space. Nearness to the diseased group triggers a flag

The phrase captures a powerful pedagogical and analytical approach: instead of starting with open sets or metrics, we begin with a primitive notion of proximity (nearness) and its dual, apartness (farness). This perspective not only clarifies classical topology but also unlocks powerful applications in data analysis, digital imaging, sensor networks, and even cognitive science.

From digital imaging to wireless networks, from clustering algorithms to quantum gravity, the near/far paradigm provides a unified, intuitive, and powerful framework. By taking proximity as primitive, we not only deepen our understanding of classical results but also open doors to new technologies and scientific insights.