  For an orbifold M we define a homology group, called t-singular homology
group t-H_q(M), which depends not only on the topological structure of the
underlying space of M, but also on the orbifold structure of M. We prove that
it is a b-homotopy invariant of orbifolds. If M is a manifold, t-H_q(M)
coincides with the usual singular homology group. We calculate the t-singular
homology groups with Z-coefficients of several orbifolds. The t-singular
homology group with rational coefficients of an orbifold M is isomorphic to the
singular homology group with rational coefficients of M (and to that of |M|).
  For each q we define a homomorphism from the orbifold homotopy group
\pi_q(M,x_0) to t-H_q(M) by using of t-modifications. Especially, t-H_1(M) is
isomorphic to the abelianization of \pi_1(M,x_0) if M is arcwise connected.
