Rigid graphs of maps

In this note we construct maps between metric separable connected spaces X and Y such that the graphs are connected, dense and rigid subspaces o f the Cartesian product X x Y . From this result it follows that there is no maximal topology among metric separable connected topologies on a given set X. In this note we shall construct maps between metric separable connect spaces X and Y such that the graphs are connected, dense and rigid subspaces of the Cartesian product X x Y . The first construction of a map f : R * R with the connected and dense graph in the plane and satisfying the Cauchy equation f ( x ) + f ( y ) = f ( x + y ) was given by F.B. Jones [3] in 1942. More general construction one can find in [4]. In order to obtain the existence o f rigid graphs o f maps, we shall utilize, in the proof, an idea o f W. Sierpiński from [5]. A similar method is also used in de G root’s paper [2]. Spaces considered here are assumed to be separable and metric, i.e. we assume that they are subspaces of the Hilbert’s cube I0’. A continuous map / : X -» Y, I , 7 c / “, is called a continuous displacement [2], iff there exists a subset V<=X such that \ f (V)\ = T and F n / ( F ) = 0 , Let us notice that each homeomorphism f : X -*■ X different from the identity map, and where A' is a connected subspace of I 03, ia a continuous displacement. Indeed, s i n c e i d * , there exists a point x e X such that f ( x ) ^ x . Choose disjoint open sets V, W<= X such that x e V and f ( x ) e f ( V ) c. W. Since X is a connected metric space hence \V \= 2 C0. Thus, 1 /(101=2" and V n f ( V ) = 0 . For more exhaustive information on continuous displacements, the reader can refer to de G root’s paper [2]. Received April 04, 1983. AMS (MOS) subject classification (1980). Primary 54C08. * Instytut Matematyki Uniwersytetu Śląskiego, Katowice, ul. Bankowa 14, Poland