Extraordinary Homology
If one defines a homology theory axiomatically (via the Eilenberg-Steenrod axioms), and then relaxes one of the axioms (the dimension axiom), one obtains a generalized theory, called an extraordinary homology theory. These originally arose in the form of extraordinary cohomology theories, namely K-theory and cobordism theory. In this context, singular homology is referred to as ordinary homology.
Read more about this topic: Singular Homology