By using separated sets and spanning sets, some equivalent definitions of topological pressure of a semigroup of continuous maps are given, and several of their basic properties are provided. We also answer an open problem of Bis and Urbanski[3]. That is, letting fi, i=2,…, k, be homeomorphisms acting on a compact metric space, Gi={idx,f2, G11={idx,f21,…,fk1} and letting G and G1denote the semigroups generated by G1and G1respectively, we give an example showing that the topological entropy of G does not equal the topological entropy of G1.In Chapter1, we introduce the development history and the investigation status of topological entropy and topological pressure.Some general conceptions, knowledge and notations are given in preliminaries in Chapter2.In section3, we give the proof of a theorem in [23].In section4, we give some equivalent definitions of the topological pressure.In section5, we give some properties of the topological pressure.In section6, we provide an example which answers the open problem in [3]. This example also shows that there exist properties of the topological pressure of a single transformation which are not inherited by the case of semigroup of continuous maps.
