Department of Mathematics, Faculty of Science, University of Qom, P.O.Box 3716146611, Qom, Iran.
Let $X$ be an infinite set, equipped with a topology $\tau$. In this paper we studied the relationship between $\tau$, and ultrafilters on $X$. We can discovered, among other thing, some relations of the Robinson's compactness theorem, continuity and the separation axioms. It is important also, aspects of communication between mathematical concepts.