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General Mathematics Seminar of the St. Petersburg Division of Steklov Institute of Mathematics, Russian Academy of Sciences
February 2, 1998, St. Petersburg, POMI, room 311 (27 Fontanka)

Topologies on products and function spaces

A. A. Ivanov

Abstract: 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.