The Simple Form of Urysohn Lemma and Its Application

We present the sufficient and necessary conditions that there is continuous functions which supports is contained in certain open set and the value is constant in some closed subset of the open set. At the same time, we establish Urysohn lemma in the perfect Cover of general topological space and obtain a more general form of this theorem and construct the sufficient and necessary conditions which the sets are function separared. order preserving theory is utilized to prove a more general Uryshon Lemma and we obtain the sufficient and necessary conditions which a set family is a perfect cover. Then we survey the connection between the various Urysohn's lemmas and obtain an important property of perfect cover. Finally, we give the application of Urysohn lemma and prove the generalized Tietze expansion theorem.