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.

Since every point of a non-Hausdorff manifold has a Euclidean neighborhood, it is easy to show that every non-Hausdorff manifold is T 1 .
We now briefly review some of the well-known properties of non-Hausdorff manifolds.Since R n is locally compact, a non-Hausdorff manifold of dimension n is locally compact.In some of literature, compactness is only defined in Hausdorff spaces.In such cases, compact subsets must be closed.Compact subsets of T 1 -spaces, however, need not to be closed.This remains true for non-Hausdorff manifolds Example 1.2 .A non-Hausdorff manifold of dimension n must be locally connected.Since a non-Hausdorff manifold M has a countable base of open sets, M is Lindel öf; that is, every open cover of M has a countable subcover.Further, since locally compact Lindel öf spaces are sigma-compact, it follows that a non-Hausdorff manifold M of dimension n is sigma-compact.Finally, we note that when M is not Hausdorff, it is not regular.
We now consider the property of paracompactness.A Hausdorff space X is paracompact if every open covering U of X has a locally finite refinement V.That is, each V ∈ V is contained in some U ∈ U and each x ∈ X has a neighborhood N which meets only finitely many sets in V. Paracompactness can be defined for T 1 -spaces as follows.A T 1 -space X is paracompact if and only if each open covering of X has an open barycentric refinement, where V is a barycentric refinement of U if the collection {St x, V : x ∈ X} refines U, where St x, V ∪{V ∈ V : x ∈ V }.A space is metacompact if every open cover has a point finite refinement.Since Hausdorff second countable manifolds are metrizable, they are paracompact and hence metacompact.In 1 , an example of a non-Hausdorff manifold which is not metacompact is given.We present another one.ii For each q, 1 ∈ Q×{1}, a basic open neighborhood of q, 1 is of the form { q, 1 }∪ U \ {q} where U is an open neighborhood of q in R with the usual topology.
Claim 1.The non-Hausdorff manifold M is not metacompact.
By the way U is defined, no element of U contains more than one element of Q × {1}.Since V is a refinement of U, no element of V contains more than one element of Q × {1}.Hence, V j / V k whenever j / k.By Cantor's Intersection theorem, there exists Remark 1.3.In the above example, 0, 1 is compact and Hausdorff but not closed.
Remark 1.4.For each n ∈ N, R n is a complete metric space and Q n is a countable dense subset of R n .Therefore, a construction similar to the one above can be used to create a non-Hausdorff manifold of dimension n that is not metacompact.

Compatible Apparition Points
If a manifold M of dimension n is non-Hausdorff, there exist at least two points x and y which cannot be separated by disjoint open sets.Also, the points x and y cannot be contained in the same Euclidean neighborhood since Euclidean neighborhoods are Hausdorff.
Definition 2.1.Let M be a non-Hausdorff manifold and let x and y be distinct points of M. Then x and y are compatible apparition points if there do not exist disjoint open sets U and V with x ∈ U and y ∈ V .By a "set of compatible apparition points," we will mean that any pair of distinct points in the set are compatible apparition points.
Remark 2.2.Since a non-Hausdorff manifold is locally Hausdorff, then no more than one element of a set of compatible apparition points can be contained in a single Euclidean neighborhood.Hence, a set of compatible apparition points is a closed discrete set.
Remark 2.3.Since a non-Hausdorff manifold has a countable base and each point is contained in its own Euclidean neighborhood, any set of compatible apparition points must be countable.
A non-Hausdorff manifold can have an uncountable collection of sets of compatible apparition points.ii For each Note that for each x ∈ C, {x, x, 0 } is a set of compatible apparition points.Also, note that since each ε can be chosen to be rational, X is second countable.
Recall that a subset of a topological space is nowhere dense if the interior of its closure is empty.Proposition 2.5.Let S be a set of compatible apparition points in a non-Hausdorff manifold M. Then S is nowhere dense in M.
Proof.Since S is closed and discrete and every element of M has a Euclidean neighborhood, S is the frontier of M \ S which is open.Hence, S is nowhere dense by 2, 4G part 2 on page 37 .
Proposition 2.6.Let M be an n-dimensional non-Hausdorff manifold.Suppose that M contains a nonempty set S of compatible apparition points.Then every continuous function from M to a Hausdorff space X is constant on S.
Proof.Suppose that f : M → X is continuous.Attempting a contradiction, suppose that Theorem 2.7.In a non-Hausdorff manifold, the set of points which are not apparition points is dense.
Proof.Suppose that M is a non-Hausdorff manifold.Since M is locally Hausdorff, Lemma 4.2 of 3 implies that each x ∈ M has a dense open Hausdorff neighborhood U x .Since M is Lindel öf, the cover {U x } x∈M has a countable subcover C. Since M is Baire, ∩C is dense in M. Since the elements of C are Hausdorff, any point in ∩C can be separated from any other point in M. Therefore, ∩C is a dense set of nonapparition points.

Example 1 . 2 .
A non-Hausdorff manifold M need not to be metacompact.Let M R ∪ Q × {1} and define a topology on M as follows.i For each x ∈ R, a basic open neighborhood of x is open in R with the usual topology.

Example 2 . 4 .
Let C denote the Cantor ternary set and define X R ∪ C × {0} .Define a topology on X as follows.i For each x ∈ R, a basic open neighborhood of x is open in R with the usual topology.