It is easy to see that if $A\times B$ is homeomorphic to $A\times C$ for topological spaces $A$, $B$, $C$, then one may not conclude that $B$ and $C$ are homeomorphic (for example, take $C=B^2$, $A=B^∞$). The question is: for which $A$ such conclusion is true?
Witold Rosicki has a lot of results of this sort (usually under some conditions on $B$ and $C$). For instance,
http://matwbn.icm.edu.pl/ksiazki/cm/cm72/cm7217.pdf
http://dx.doi.org/10.1016/j.topol.2004.02.013
http://dx.doi.org/10.1016/j.topol.2003.04.002
There also exist papers of a different flavor on this subject