The fixed-point property for represented spaces

We investigate which represented spaces enjoy the fixed-point property, is property that every continuous multi-valued function has a fixed-point. study basic theory of this notion and its uniform version. provide complete characterization countable-based T0-spaces with showing they are exactly pointed ω-continuous dcpos. prove whose lattice open sets enjoys countably-based spaces. While role played by free functions in diagonal argument well-known, we show how it can be adapted to functions, apply technique identify base-complexity Kleene-Kreisel spaces, was an problem.