point-set topology

Point-set topology is a branch of mathematics focused on the properties and theory of topological spaces and metric spaces, with an emphasis on set theory. It deals with concepts like closeness and limits, defined by relationships between sets rather than distances.12

A topological space is defined by a set of points and a collection of subsets called open sets. These open sets must satisfy specific rules: the empty set and the entire set are included, the intersection of any finite number of open sets is open, and the union of any collection of open sets is open. A key concept is the limit point: a point is a limit point of a set if every open set containing it also contains other points from that set.32

Topology, often called "rubber sheet geometry," studies properties that remain unchanged during continuous deformations like stretching or bending, without tearing or gluing. This differs from geometry, which often focuses on measurable quantities like length or angles.4