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What is a Complex in Topology? 📂Topological Data Analysis

What is a Complex in Topology?

Overview

In mathematics, the term “Complex” usually refers to complex numbers, but in geometry or topology, “Complex” signifies something like the following terms.

Terminology

A Complex is made up of topologically simple $S$s, whose intersections are of a lower dimension but of the same kind as $S$.

Description

As this ambiguous expression suggests, there isn’t exactly a ‘definition’.

Whether we call the simple things a Simplex or refer to Complex as a Complex, minor details are not our focus. When we mention “Complex” in this context, one can just assume that’s what is being referred to and move on. A Complex can vary significantly depending on the method, and understanding its types becomes natural after seeing a few specific examples.