Compactification of the Stationary Channel Space

Topologization of the set consisting of all channels is given. This result is applied to the proof that the entropy rates and the transmission rate are jointly upper semi-continuous on the product topological space constructed from the stationary regular Borel probability measures and the stationary channels. Finally, it is proved that there exists a combination of an input stationary measure and a stationary channel having finite memory which can make the transmission rate maximal