The abstract of the talk (see below) may be downloaded in one of the following formats:

Direct products are defined in arbitrary categories and are unique,
whenever they exist. In the category H(Top) of topological
spaces and homotopy classes of mappings, for any two spaces $X,Y$
their product exists and is given by the Cartesian product $X\times Y$ and
by the homotopy classes $[\pi_X]$ and $[\pi_Y]$ of the canonical projections
$\pi_X\colon X\times Y\to X$, $\pi_Y\colon X\times Y\to Y$, respectively.
Since shape theory is a modification of homotopy theory, it is natural
to ask whether in the shape category
Sh(Top) the Cartesian product $X\times Y$ and the shape
morphisms $S[\pi_X], S[\pi_y]$, induced by $[\pi_X]$ and
$[\pi_Y]$, form the direct product of $X$ and $Y$ ? In 1974 J.E. Keesling [1]
exhibited a simple
(non-compact) space $X\subseteq\R^2$ such that $X\times X$ and the two shape
morphisms $S[\pi_X]\colon X\times X\to X$ do not form a
product in Sh(Top). However, he gave a positive answer in the case when
$X$ and $Y$ are compact Hausdorff spaces. In 1977 Y. Kodama
proved that the answer is positive also in the case when $X$ is an FANR
(fundamental absolute neighorhood retract) and $Y$ is a paracompact space~[2].

The main aim of the present talk is to announce analogous positive results
in strong shape [4], i.e. for the strong shape category SSh(Top) and
the strong shape functor $\overline{S}\colon{\rm
H(Top)}\to {\rm SSh(Top)}$  (for definitions see [3]).\medskip

\begin{thm}
If $X$ and $Y$ are
compact Hausdorff spaces, then $X\times Y$, $\overline{S}[\pi_X]$ and
$\overline{S}[\pi_Y]$ form the product of $X$ and $Y$ in {\rm
SSh(Top)}.
\end{thm}

\begin{thm}
If $X$ is an {\rm
FANR} and $Y$ is a finite-dimensional space $($more general, a finitistic
space$)$, then $X\times Y$, $\overline{S}[\pi_X]$ and
$\overline{S}[\pi_Y]$ form the product of $X$ and $Y$ in {\rm
SSh(Top)}.
\end{thm}

\smallskip
\begin{thebibliography}{10}
\bibitem{Kees74}
J.\,E.~Keesling, {\em Products in the shape category
and some applications,} Symp.\ Math.\ Istituto Nazionale di Alta
Matematica {\bf16} (1973), Academic Press, New York, 1974, 133--142.

\bibitem{Koda77}
Y.~Kodama, {\em On product of shape and a question of
Sher,} Pacific J.~Math.{} {\bf72} (1977), 115--134.

\bibitem{Mard2000}
S.~Marde\v{s}i\'c,
{\em Strong shape and homology,} Springer Monographs in Mathematics,
Springer, Berlin 2000.

\bibitem{Mard2001}
S. Marde\v si\'c,
{\em Strong expansions of products and
products in strong shape,} (submitted).

\end{thebibliography}

\vfill\noindent
{\it Mathematics Subject Classification\/} 54B35, 54C56, 55P55.\\[10pt]
{\it Key words and phrases\/}: inverse limit, strong expansion, homotopy
expansion, direct product, shape, strong shape, finitistic space.

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