Singular homology and cohomology as fundamental algebraic invariants in the classification of topological spaces

To gain insight into topological spaces, algebraic topology attempts to give topological spaces algebraic structures, which are preserved during continuous deformation. Homology and cohomology theories are among the most powerful and long-standing constructions in the field, which provide constructions of topological data worldwide in an algebraic form. These theories began with the foundational work of Henri Poincaré to become a foundational pillar of current mathematics with extensive links to homological algebra, differential geometry, algebraic geometry, and number theory. The paper gives an exposition of singular homology and cohomology with a lot of detail and rigor with emphasis on the construction of chain complexes, functoriality and exact sequences. We build the theory in a systematic manner starting with singular simplices and operator of a boundary, then relative homology, excision theory and Mayer Vincent Theory, and then finally cohomology Rings, universal coefficient theorems and Poincaré duality. We have, throughout, indicated the conceptual importance of these results and their place in current mathematical research. The article will be a complete and stand-alone source of information that can be published in peer-reviewed journals.