Topologies on products and function spaces

There are a surprising connections between topological structures on products $X\times Y$ and topological structures on spaces of functions (mappings) $Y/Z$ — in other definitions $C(Y,Z)$, $Z^Y$. Not going into details we say that for a topological structure $T$ on $X/Y$ there exists the corresponding (conjugate) topological structure $T^*$ on $Y/Z$ and for a topological structure $T$ on $Y/Z$ there exists the corresponding (conjugate) topological structure $T_*$ on $X\times Y$. If $(T_*)^*=T$ ($(T^*)_*=T$), then the topological structures $T$ and $T_*$ ($T$ and $T^*$) are called dual ones. For example, the usual topology on $X\times Y$ and the compact-open topology on $Y/Z$ are dual, the topology of pointwise convergence on $Y/Z$ and the topology on $X\times Y$ defined by convergencies of directed systems of points stationary for some co-ordinate are dual too.