I'm having trouble understanding a certain situation having to do with (topological) intersection numbers of submanifolds of a manifold with boundary. This whole question will be based on theorem 11.9 in chapter 6 of "Topology and Geometry" by Bredon. This is solely due to the fact that I haven't been able to find any other reference that develops topological intersection theory on a manifold with boundary (suggestions for such references are most welcome).
Anyway, the theorem (also quoted more fully in this question) states that given a smooth manifold with boundary $W$, and $N, K $ two smooth manifolds with boundary embedded in $W$, such that $N$ meets $K$ transversely in $W - \partial W$, and $N$ meets $\partial W$ transversely in $\partial N$, and $K$ meets $\partial W$ transversely in $\partial K$, then it follows that $[K\cap N]_W = [N]_W \bullet [K]_W$, where $[M]$ means the homology class of $M$ in $H_i(W, \partial W)$, and $\bullet$ is the intersection product.
In words, the intersection product of the homology classes of $N, K$, in the reduced homology group mentioned above, is the homology class of their set-theoretic intersection.
Now consider the following situation: let $W$ be the square $[0,1]\times [0,1]$. This is a manifold with boundary; to make it smooth, round the corners slightly (this is irrelevant for the example). Also let (where $[(a,b),(c,d)]$ means the straight line segment from $(a,b)$ to $(c,d)$)
$N = [(0.2, 0),(0.2, 1)]$
$N' = [(0.2, 0),(0.2, 0.5)] \cup [(0.2, 0.5),(0.8, 0.5)] \cup [(0.8, 0.5),(0.8, 1)]$
$K = [(0.5, 0),(0.5, 1)]$
Again, technically $N'$ is not smoothly embedded in $W$, but round its corners slightly to make it smooth (this is irrelevant to the example anyway).
Now clearly $N, K$ satisfy the transversality conditions of the theorem. Their set-theoretic intersection is empty. And $N', K$ also satisfy the transversality conditions; but their set-theoretic intersection is nonempty, and consists of the point $(0.5, 0.5)$. The problem with this is that $N, N'$ are clearly homologous to each other as cycles in $H_i(W, \partial W)$, because their difference is homologous to 0 in that group (their difference is the three sides of a rectangle whose fourth side is in $\partial W$. In particular that rectangle is homotopic to a point). This means that according to the theorem above, $[K\cap N]_W$ and $[K\cap N']_W$ should agree, because they are both equal to $[N]_W \bullet [K]_W = [N']_W \bullet [K]_W$ (this equality follows because $N$ and $N'$ are homologous). But they seem to not agree, since one of them is the homology class of the empty cycle and the other is the homology class of a single point. What am I missing?