topological space
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AI overview
A topological space is a mathematical structure consisting of a set of points along with a defined collection of neighborhoods around those points that follow specific rules. This framework matters because it provides a general way to study properties like continuity and connectedness that don't depend on measuring distances, making it useful across many areas of mathematics.
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Article
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points, along with an additional structure called a topology, which can be defined as a set of neighbourhoods for each point that satisfy some axioms formalizing the concept of closeness. There are several equivalent definitions of a topology, the most commonly used of which is the definition through open sets.
A topological space is the most general type of a mathematical space that allows for the definition of limits, continuity, and connectedness. Common types of topological spaces include Euclidean spaces, metric spaces and manifolds.