Brown representability theorem

Brown’s representability theorem is a fundamental result in homotopy theory stating that, under certain conditions, any cohomology theory (which assigns algebraic invariants to spaces) can be represented by a topological space. In simpler terms, it means that for a broad class of invariants used to study shapes, there exists a specific space such that measuring that space’s properties captures the invariant completely. This bridges the abstract algebraic aspects of topology with concrete geometric objects, enabling mathematicians to study and classify spaces through representative spaces linked to their invariants.