In thie lecture we define a new homology theory, called Čech cohomology. Rather than working in maximal generality, we give the definition only in the special case that we need.

For this we introduce a Euclidean neighbourhood retract, and define the Čech cohomology only for a pair $K \subset L$ of compact sets in a given Euclidean neighbourhood retract. These groups are written $\check{H}^{\bullet}(L,K)$.

Next lecture we will relate Čech cohomology for compact sets of a manifold, via the following Duality Theorem:

Let $M$ be an $n$-dimensional oriented topological manifold. Then for every pair $K \subset L$ of compact subsets of $M$, one has an isomorphism
$$\check{H}^k(L,K) \cong H_{n-k}(M \setminus K, M \setminus L).$$

The famous Poincaré Duality Theorem is a special case of the above result (simply take $M$ to be closed, take $L = M$ and $K = \emptyset$.)