One can consider various spaces of continuous transitive maps of a closed interval onto itself. They can consist of all transitive maps, piecewise monotone transitive maps, piecewise linear transitive maps, etc. Moreover, we can restrict our attention to piecewise monotone maps with a given number of pieces, or various unions of such spaces. We consider all those spaces with the topology of uniform convergence. In such a way we get many topological spaces, and then we can ask about their properties. We give some answers, concentrating on properties such as arcwise connectedness and contractibility.

This is joint work with Sergiy Kolyada and Lubomir Snoha.