Well known

if X is a compact space and if some sequence{fn}is a sequence ofcontinuous real value functions separates points on X

then X is metrizable.In this paper

first we extended this conclusion from compact space to loca lly compact and paracompact one.Second

the equivalance of the metrizabilltyand existence of this{fn} is verified on σ-locally compact space.And thisresult is applied to topological group

we obtain a conclusion as follows:Suppose X is a σ-locally compact group then X satisfies the secondaxiom of countability if and oniy if there is a sequence of the characters {ra}

which separates points on X.