Today we finally got round to defining singular homology, which is a collection of functors

$$H_n \colon \mathsf{Top} \to \mathsf{Ab}, \qquad n \ge 0.$$

It will take us sometime before we can actually prove anything interesting about singular homology (as evidenced by the fact that the most exciting question on Problem Sheet D asks you to prove that the singular homology of a point is zero for n>0 .)

Next lecture we will show that—just like with $\pi_1$—actually $H_n$  induces a functor $\mathsf{hTop} \to \mathsf{Ab}$. The true power of singular homology won't become apparent until we study excision in Lecture 14.