On almost fixed point theorems and fixed point theorems on topological spaces

The concept of R-subset of a topological space is introduced,KKM type theorem on general topological space is obtained by using the openversion of classical KKM principle,and continuous selection theorems are established,then almost fixed point theorems for uppersemicontinuous multimap are obtained and fixed point theorems for upper semicontinuous multimap with closed values are given on locally uniform spaces.These results generalize and improve many corresponding conclusions.