Fundamental class of a manifold

Definition
Let $$M$$ be a compact connected orientable manifold of dimension $$n$$ and $$R$$ a commutative ring. A fundamental class for $$M$$ is an element of $$H_n(M;R)$$ whose image in $$H_n(M,M\setminus x;R)$$ is a generator (note that $$H_n(M, M \setminus x;R)$$ is clearly a free $$R$$-module of rank $$1$$).

A fundamental class corresponds to a choice of orientation.

Note that if $$M$$ is not compact or if $$M$$ is not orientable, no fundamental class exists. If $$M$$ is a union of connected components each of which is compact and orientable, we can mimic the above definition by defining a fundamental class of $$M$$ as a sum of fundamental classes of each of its connected components; however, the term fundamental class is typically reserved for connected manifolds.