A Note on Topological Properties of Non-Hausdorff Manifolds

The notion of compatible apparition points is introduced for non-Hausdorff manifolds, and properties of these points are studied. It is well known that the Hausdorff property is independent of the other conditions given in the standard definition of a topological manifold. In much of literature, a topological manifold of dimension n is a Hausdorff topological space which has a countable base of open sets and is locally Euclidean of dimension n. We begin with the definition of a non-Hausdorff topological manifold.