Borel–Moore homology

Borel–Moore homology is a mathematical tool used to study the shape and structure of spaces, especially those that are not compact or have infinite extent. Unlike standard homology, which focuses on finite or closed features, Borel–Moore homology accounts for 'locally finite' chains, allowing for the analysis of non-compact spaces by considering infinite but controlled chains. It provides a way to understand the topology of open, non-compact, or singular spaces by capturing their essential features at infinity, enhancing our ability to analyze complex geometrical and topological structures in areas like algebraic geometry and geometric topology.