ITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS – N.42–2019 (744–755) 744 Urysohn lemma in semi-linear uniform spaces Amani Rawshdeh Department of Mathematics Al- Balqa‘ Applied University Alsalt Jordan amanirawshdeh@bau.edu.jo Suad Alhihi ∗ Department of Mathematics Al- Balqa‘ Applied University Alsalt Jordan suadhihi@bau.edu.jo Abstract. In topology Urysohn Lemma is widely applicable, where it is commonly used to construct continuous functions with various properties on normal space. In this paper we shall present Urysohn Lemma in semi linear uniform spaces, besides we shall give a characterization of the closure in semi-linear uniform space, then we shall use this characterization to answer the question which given in [12], by A.Tallafha and R. Khalil namely (If ρ(x, A)=Δ, must x ∈ A l ). Keywords: uniform spaces, Semi-linear uniform spaces, topological spaces, metric spaces, types of metric spaces. 1. Introduction One of the most important generalizations of a metric spaces is uniform space. Uniform spaces is a concept lies between metric spaces and topological struc- ture, that is used to define uniform properties such as completeness, uniform continuity and uniform convergence. The uniform spaces have been studied extensively through years. The notion of uniformity has been investigated by several mathematician such as Weil [18] ,[19] , and [20] . L.W. Cohen [4] , and [5] . Graves [7]. The theory of uniform spaces was given by Burbaki in [3] . Also Wiel’s in his booklet [20] , defined the notion of uniformly continuous mapping. In 2009, the notion of a uniform space led A. Tallafha and R. Khalil to define a beautiful space which is a mixture of analysis and topology, namely semi- linear uniform space [12], also they studied some cases of best approximation in such spaces, besides they defined a set valued map ρ, called metric type, on semi-linear uniform spaces that enables one to study analytical concepts on semi-linear uniform space. *. Corresponding author