Pseudonormal spaces

When two disjoint closed sets in a topological space cannot be separated by disjoint open sets, the space fails to be a normal space. When one of the two closed sets is countable, the space fails to satisfy a weaker property than normality. A space X is said to be a pseudonormal space if H and K can always be separated by two disjoint open sets whenever H and K are disjoint closed subsets of X and one of them is countable. In this post, we discuss several non-normal spaces that actually fail to be pseudonormal. We also give an example of a pseudonormal space that is not normal.

We work with spaces that are at minimum T_1 spaces, i.e., spaces in which singleton sets are closed. Then any pseudonormal space is regular. To see this, let X be T_1 and pseudonormal. For any closed subset C of X and for any point x \in X-C, we can always separate the disjoint closed sets \left\{ x \right\} and C by disjoint open sets. This is one reason why we insist on having T_1 separation axiom as a starting point. We now show some examples of spaces that fail to be pseudonormal.

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Some Non-Pseudonormal Examples

All three examples in this section are spaces where the failure of normality is exhibited by the inability of separating a countable closed set and another disjoint closed set.

Example 1
This example of a non-normal space that fails to be pseudonormal is defined in the previous post called An Example of a Completely Regular Space that is not Normal. This is an example of a Hausdorff, locally compact, zero-dimensional (having a base consisting of closed and open sets), metacompact, completely regular space that is not normal. We state the definition of the space and present a proof that it is not pseudonormal.

Let E be the set of all points (x,y) \in \mathbb{R} \times \mathbb{R} such that y \ge 0. For each real number x, define the following sets:

    V_x=\left\{(x,y) \in E: 0 \le y \le 2 \right\}

    D_x=\left\{(s,s-x) \in E: x \le s \le x+2 \right\}

    O_x=V_x \cup D_x

The set V_x is the vertical line of height 2 at the point (x,0). The set D_x is the line originating at (x,0) and going in the Northeast direction reaching the same vertical height as V_x as shown in the following figure.

The topology on E is defined by the following:

  • Each point (x,y) \in E where y>0 is isolated.
  • For each point (x,0) \in E, a basic open set is of the form O_x - F where (x,0) \notin F and F is a finite subset of O_x.

The x-axis in this example is a closed and discrete set of cardinality continuum. Amy two disjoint subsets of the x-axis are disjoint closed sets. The two closed sets that cannot be separated are:

    H=\left\{(x,0) \in E: x \text{ is rational} \right\}

    K=\left\{(x,0) \in E: x \text{ is irrational} \right\}

For each (x,0), let W_x=O_x-F_x where F_x \subset O_x is finite and (x,0) \notin F_x. Furthermore, break up F_x by letting F_{x,d}=F_x \cap D_x and F_{x,v}=F_x \cap V_x. Let U and V be defined by:

    U_H=\bigcup \limits_{(x,0) \in H} W_x

    U_K=\bigcup \limits_{(x,0) \in K} W_x

The open sets U_H and U_K are essentially arbitrary open sets containing H and K respectively. We claims that U_H \cap U_K \ne \varnothing.

Define the projection map \tau_1:\mathbb{R}^2 \rightarrow \mathbb{R} by \tau_1(x,y)=x. Let A and B be defined by:

    A=\bigcup \left\{\tau_1(F_{x,d}): (x,0) \in H \right\}

    B=\left\{(x,0) \in K: (x,0) \notin A \right\}

The set A is countable. So the set B is uncountable. Choose (x,0) \in B. Choose (a,0) \in H on the left of (x,0) and close enough to (x,0) such that V_x \cap D_a=\left\{t \right\} and t \notin F_{x,v}. This means that

    t \in V_x \cup D_x -F_x=O_x-F_x=W_x

    t \in V_a \cup D_a -F_a=O_a-F_a=W_a.

Thus U_H \cap U_K \ne \varnothing. We have shown that the space E is not pseudonormal and thus not normal.

Example 2
The Sorgenfrey line is the real line \mathbb{R} topologized by the base consisting of half open and half closed intervals of the form [a,b)=\left\{x \in \mathbb{R}: a \le x < b \right\}. In this post, we use S to denote the real line \mathbb{R} with this topology.

The Sorgenfrey line S is a classic example of a normal space whose square S \times S is not normal. In the Sorgenfrey plane S \times S, the set \left\{(x,-x) \in S \times S: x \in \mathbb{R} \right\} is a closed and discrete set and is called the anti-diagonal. The proof presented in this previous post shows that the following two disjoint closed subsets of S \times S

    H=\left\{(x,-x) \in S \times S: x \text{ is rational} \right\}

    K=\left\{(x,-x) \in S \times S: x \text{ is irrational} \right\}

cannot be separated by disjoint open sets. The argument is based on the fact that the real line with the usual topology is of second category. The key point in the argument is that the set of the irrationals cannot be the union of countably many closed and nowhere dense sets (in the usual topology of the real line).

Thus S \times S fails to be pseudonormal. This example shows that normality can fail to be preserved by taking Cartesian product in such a way that even pseudonormality cannot be achieved in the Cartesian product!

Example 3
Another example of a non-normal space that fails to be pseudonormal is the Niemmytzkis’ plane (Example 2 in in this previous post). The underlying set is N=\left\{(x,y) \in \mathbb{R} \times \mathbb{R}: y \ge 0 \right\}. The points lying above the x-axis have the usual Euclidean open neighborhoods. A point (x,0) in the x-axis has as neighborhoods \left\{(x,0) \right\} together with the interior of a disc in the upper half plane that is tangent at the point (x,0). Consider the following the two disjoint closed sets on the x-axis:

    H=\left\{(x,0): x \text{ is rational} \right\}

    K=\left\{(x,0): x \text{ is irrational} \right\}

The disjoint closed sets H and K cannot be separated by disjoint open sets (see Niemytzki’s Tangent Disc Topology in [2], Example 82). Like Example 2 above, the argument that H and K cannot be separated is also a Baire category argument.

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An Example of Pseudonormal but not Normal

Example 4
One way to find such a space is to look for spaces that are non-normal and see which one is pseudonormal. On the other hand, in a pseudonormal space, countable closed sets are easily separated from other disjoint closed sets. One space in which “countable” is nice is the first uncountable ordinal \omega_1 with the order topology. But \omega_1 is normal. So we look at the Cartesian product \omega_1 \times (\omega_1 +1). The second factor is the successor ordinal to \omega_1 or as a space that is obtained by tagging one more point to \omega_1 that is considered greater than all the points in \omega_1. Let’s use X \times Y=\omega_1 \times (\omega_1 +1) to denote this space.

The space X \times Y is not normal (shown in this previous post). In the previous post, X \times Y is presented as an example showing that the product of a normal space with a compact space needs not be normal. However, in this case at least, the product is pseudonormal.

Let \alpha < \omega_1. Then the square \alpha \times \alpha as a subspace of X \times Y is a countable space and a first countable space. So it has a countable base (second countable) and thus metrizable, and in particular normal. Any countable subset of X \times Y is contained in one of these countable squares, making it easy to separate a countable closed set from another closed set.

Let H and K be disjoint closed sets in X \times Y such that H is countable. Then there is some successor ordinal \mu < \omega_1 (\mu=\alpha+1 for some ordinal \alpha<\omega_1) such that H \subset \mu \times \mu. Based on the discussion in the preceding paragraph, there are disjoint open sets O_H and O_K in \mu \times \mu such that H \subset O_H and (K \cap (\mu \times \mu)) \subset O_K. With \mu being a successor ordinal, the square \mu \times \mu is both closed and open in X \times Y. Then the following sets

    V_H=O_H

    V_K=O_K \cup (X \times Y-\mu \times \mu)

are disjoint open sets in X \times Y separating H and K.

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Some Comments about Examples 1 – 3

In each of Examples 1, 2 and 3 discussed above, there is a closed and discrete set of cardinality continuum (the x-axis in Examples 1 and 3 and the anti-diagonal in Example 2). So the extent of each of these three spaces is continuum. Note that the extent of a space is the maximum cardinality of a closed and discrete subset.

In each of these examples, it just so happens that it is not possible to separate the rationals from the irrationals in the x-axis or the anti-diagonal by disjoint open sets, making each example not only not normal but also not pseudonormal.

What if we consider a smaller subset of the x-axis or anti-diagonal? For example, consider an uncountable set of cardinality less than continuum. Then what can we say about the pseudonormality or normality of the resulting subspaces? For Example 1, the picture is clear cut.

In Example 1, the argument that H and K cannot be separated is a “countable vs. uncountable” argument. The argument will work as long as H is a countable dense set in the x-axis (dense in the usual topology) and K is any uncountable set.

For Example 2 and Example 3, the argument that H and K cannot be separated is not a “countable vs. uncountable” argument and instead is a Baire category argument. The fact that one of the closed sets is the irrationals is a crucial point. On the other hand, both Example 2 and Example 3 (especially Example 3) are set-theoretic sensitive examples. For Example 2 and Example 3, the normality of the resulting smaller subspaces is dependent on some extra axioms beyond ZFC. For pseudonormality, it could be set-theoretic sensitive too. We give some indication here why this is so.

Let S be the Sorgenfrey line as in Example 2 above. Assuming Martin’s Axiom and the negation of the continuum hypothesis (abbreviated by MA + not CH), for any uncountable X \subset S with \lvert X \lvert < c, X \times X is normal but not paracompact (see Example 6.3 in [1] and see [3]). Even though X \times X is not exactly a comparable example, this example shows that restricting to a smaller subset on the anti-diagonal seems to make the space normal.

Example 3 has an illustrious history with respect to the normal Moore space conjecture. There is not surprise that extra set-theory axioms are used. For any subset B of the x-axis, let N(B) be the space defined as in Example 3 above except that only points of B are used on the x-axis. Assuming MA + not CH, for any uncountable B that is of cardinality less than continuum, it can be shown that N(B) is normal non-metrizable Moore space (see Example F in [4]). So by assuming extra axiom of MA + not CH, we cannot get a non-pseudonormal example out of Example 3 by restricting to a smaller uncountable subset of the x-axis. Under other set-theoretic axioms, there exists no normal non-metrizable Moore space. Just because this is a set-theoretic sensitive example, it is conceivable that N(B) could be a space that is not pseudonormal under some other axioms.

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Reference

  1. Burke, D. K., Covering Properties, Handbook of Set-Theoretic Topology (K. Kunen and J. E. Vaughan, eds), Elsevier Science Publishers B. V., Amsterdam, 347-422, 1984.
  2. Steen, L. A., Seebach, J. A., Counterexamples in Topology, Dover Publications, Inc, Amsterdam, New York, 1995.
  3. Przymusinski, T. C., A Lindelof space X such that X \times X is normal but not paracompact, Fund. Math., 91, 161-165, 1973.
  4. Tall, F. D., Normality versus Collectionwise Normality, Handbook of Set-Theoretic Topology (K. Kunen and J. E. Vaughan, eds), Elsevier Science Publishers B. V., Amsterdam, 685-732, 1984.

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\copyright \ 2014 \text{ by Dan Ma}

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