Topological half-exact functor
= Topological half-exact functor
= Half-exact functor
{synonym}
For a contravariant pointed <functor> on based CW <homotopy> types, half-exactness on a <cofibration> means that $F(X/A)\to F(X)\to F(A)$ is exact at $F(X)$: the image is the inverse image of the distinguished point. This does not demand injectivity or surjectivity at the ends. Set-valued exactness and group-valued exactness must be distinguished.