Bing’s Example G

Bing’s Example G is an example of a topological space that is normal but not collectionwise normal. It was introduced in an influential paper of R. H. Bing in 1951 (see [1]). This paper has a metrization theorem that is now called Bing’s metrization theorem (any regular space is metrizable if and only if it has a \sigma-discrete base). The paper also introduced the notion of collectionwise normality and discussed the roles it plays in metrization theory (e.g. a Moore space is metrizable if and only if it is collectionwise normal). Example G was an influential example from an influential paper. It became the basis of construction for many other counterexamples (see [5] for one example). Investigations were also conducted by looking at various covering properties among subspaces of Example G (see [2] and [4] are two examples).

In this post we prove some basic results about Bing’s Example G. Some of the results we prove are found in Bing’s 1951 paper. The other results shown here are usually mentioned without proof in various places in the literature.

____________________________________________________________________

Bing’s Example G – Definition

Let P be any uncountable set. Let Q be the set of all subsets of P. Let F=2^Q be the set of all functions f: Q \rightarrow 2=\left\{0,1 \right\}. Another notation for 2^Q is the Cartesian product \prod \limits_{q \in Q} \left\{0,1 \right\}. For each p \in P, define the function f_p: Q \rightarrow 2 by the following:

    \forall q \in Q, f_p(q)=1 if p \in q and f_p(q)=0 if p \notin q

Let F_P=\left\{f_p: p \in P \right\}. Let \tau be the set of all open subsets of 2^Q in the product topology. We now consider another topology on 2^Q generated by the following base:

    \mathcal{B}=\tau \cup \left\{\left\{x \right\}: x \in F-F_P \right\}

Bing’s Example G is the set F=2^Q with the topology generated by the base \mathcal{B}. In other words, each x \in F-F_P is made an isolated point and points in F_P retain the usual product open sets.

____________________________________________________________________

Bing’s Example G – Initial Discussion

Bing’s Example G, i.e. the space F as defined above, is obtained by altering the topology of the product space of 2^{\lvert \mathcal{K} \lvert} many copies of the two-point discrete space where \mathcal{K} is the cardinality of the power set of the uncountable index set P we start with. Out of this product space, a set F_P of points is carefully chosen such that F_P has the same cardinality as P and such that F_P is relatively discrete in the product space. Points in F_P are made to retain the product topology and all points outside of F_P are declared as isolated points.

We now show that the set F_P is a discrete set in the space F. For each p \in P, let W_p be the open set defined by

    W_p=\left\{f \in F: f(\left\{p \right\})=1 \text{ and } f(P-\left\{p \right\})=0 \right\}.

It is clear that f_p is the only point of F_P belonging to W_p. Therefore, in the Example G topology, the set F_P is discrete and closed . In the section “Bing’s Example G is not Collectionwise Hausdorff” below, we show below that F_P cannot be separated by any pairwise disjoint collection of open sets.

The character at a point is the minimum cardinality of a local base at that point. The character at a point in F_P in the Example G topology agrees with the product topology. Points in F_P have character \lvert Q \lvert=2^{\lvert P \lvert}. Specifically if the starting P has cardinality \omega_1, then points in F_P have character 2^{\omega_1}. Thus Example G has large character and cannot be a Moore space (any Moore space has a countable base at every point).

____________________________________________________________________

Bing’s Example G is Normal

Let H_1 and H_2 be disjoint closed subsets of F. The easy case is that one of H_1 and H_2 is a subset of F-F_P, say H_1 \subset F-F_P. Then H_1 is a closed and open set in F. Then H_1 and F-H_1 are disjoint open sets containing H_1 and H_2, respectively. So we can assume that both H_1 \cap F_P \ne \varnothing and H_2 \cap F_P \ne \varnothing.

Let A_1=H_1 \cap F_P and A_2=H_2 \cap F_P. Let q_1=\left\{p \in P: f_p \in A_1 \right\} and q_2=\left\{p \in P: f_p \in A_2 \right\}. Define the following open sets:

    U_1=\left\{f \in F: f(q_1)=1 \text{ and } f(q_2)=0 \right\}
    U_2=\left\{f \in F: f(q_1)=0 \text{ and } f(q_2)=1 \right\}

Because H_1 \cap H_2=\varnothing, we have A_1 \subset U_1 and A_2 \subset U_2. Furthermore, U_1 \cap U_2=\varnothing. Let B_1=H_1 \cap (F-F_P) and B_2=H_2 \cap (F-F_P), which are open since they consist of isolated points. Then O_1=(U_1 \cup B_1)-H_2 and O_2=(U_2 \cup B_2)-H_1 are disjoint open subsets of F with H_1 \subset O_1 and H_2 \subset O_2.

____________________________________________________________________

Collectionwise Normal Spaces

Let X be a space. Let \mathcal{A} be a collection of subsets of X. We say \mathcal{A} is pairwise disjoint if A \cap B=\varnothing whenever A,B \in \mathcal{A} with A \ne B. We say \mathcal{A} is discrete if for each x \in X, there is an open set O containing x such that O intersects at most one set in \mathcal{A}.

The space X is said to be collectionwise normal if for every discrete collection \mathcal{D} of closed subsets fo X, there is a pairwise disjoint collection \left\{U_D: D \in \mathcal{D} \right\} of open subsets of X such that D \subset U_D for each D \in \mathcal{D}. Every paracompact space is collectionwise normal (see Theorem 5.1.18, p.305 of [3]). Thus Bing’s Example G is not paracompact.

When discrete collection of closed sets in the definition of “collectionwise normal” is replaced by discrete collection of singleton sets, the space is said to be collectionwise Hausdorff. Clearly any collectionwise normal space is collectionwise Hausdorff. Bing’s Example is actually not collectionwise Hausdorff.

____________________________________________________________________

Bing’s Example G is not Collectionwise Hausdorff

The discrete set F_P cannot be separated by disjoint open sets. For each p \in P, let O_p be an open subset of F such that p \in O_p. We show that the open sets O_p cannot be pairwise disjoint. For each p \in P, choose an open set L_p in the product topology of 2^Q such that p \in L_p \subset O_p. The product space 2^Q is a product of separable spaces, hence has the countable chain condition (CCC). Thus the open sets L_p cannot be pairwise disjoint. Thus L_t \cap L_s \ne \varnothing and O_t \cap O_s \ne \varnothing for at least two points s,t \in P.

____________________________________________________________________

Bing’s Example G is Completely Normal

The proof for showing Bing’s Example G is normal can be modified to show that it is completely normal. First some definitions. Let X be a space. Let A \subset X and B \subset X. The sets A and B are separated sets if A \cap \overline{B}=\varnothing=\overline{A} \cap B. Essentially, any two disjoint sets are separated sets if and only if none of them contains limit points (i.e. accumulation points) of the other set. A space X is said to be completely normal if for every two separated sets A and B in X, there exist disjoint open subsets U and V of X such that A \subset U and B \subset V. Any two disjoint closed sets are separated sets. Thus any completely normal space is normal. It is well known that for any regular space X, X is completely normal if and only if X is hereditarily normal. For more about completely normality, see [3] and [6].

Let H_1 \subset F and H_2 \subset F such that H_1 \cap \overline{H_2}=\varnothing=\overline{H_1} \cap H_2. We consider two cases. One is that one of H_1 and H_2 is a subset of F-F_P. The other is that both H_1 \cap F_P \ne \varnothing and H_2 \cap F_P \ne \varnothing.

The first case. Suppose H_1 \subset F-F_P. Then H_1 consists of isolated points and is an open subset of F. For each x \in H_2 \cap F_P, choose an open subset V_x of F such that x \in V_x and V_x contains no points of F_P-\left\{ x \right\} and V_x \cap \overline{H_1}=\varnothing. For each x \in H_2 \cap (F-F_P), let V_x=\left\{x \right\}. Let V be the union of all V_x where x \in H_2. Let U=H_1. Then U and V are disjoint open sets with H_1 \subset U and H_2 \subset V.

The second case. Suppose A_1=H_1 \cap F_P \ne \varnothing and A_2=H_2 \cap F_P \ne \varnothing. Let q_1=\left\{p \in P: f_p \in A_1 \right\} and q_2=\left\{p \in P: f_p \in A_2 \right\}. Define the following open sets:

    U_1=\left\{f \in F: f(q_1)=1 \text{ and } f(q_2)=0 \right\}
    U_2=\left\{f \in F: f(q_1)=0 \text{ and } f(q_2)=1 \right\}

Because H_1 \cap H_2=\varnothing, we have A_1 \subset U_1 and A_2 \subset U_2. Furthermore, U_1 \cap U_2=\varnothing. Let B_1=H_1 \cap (F-F_P) and B_2=H_2 \cap (F-F_P), which are open since they consist of isolated points. Then O_1=(U_1 \cup B_1)-\overline{H_2} and O_2=(U_2 \cup B_2)-\overline{H_1} are disjoint open subsets of F with H_1 \subset O_1 and H_2 \subset O_2.

____________________________________________________________________

Bing’s Example G is not Perfectly Normal

A space is perfectly normal if it is normal and that every closed subset is G_\delta (i.e. the intersection of countably many open subsets). The set F_P of non-isolated points is a closed set in F. We show that F_P cannot be a G_\delta-set. Before we do so, we need to appeal to a fact about the product space 2^Q.

According to the Tychonoff theorem, the product space 2^Q is a compact space since it is a product of compact spaces. On the other hand, 2^Q is a product of uncountably many factors and is thus not first countable. It is a well known fact that in a compact Hausdorff space, if a point is a G_\delta-point, then there is a countable local base at that point (i.e. the space is first countable at that point). Thus no point of the compact product space 2^Q can be a G_\delta-point. Since points of F_P retain the open sets of the product topology, no point of F_P can be a G_\delta-point in the Bing’s Example G topology.

For each p \in P, let W_p be open in F such that f_p \in W_p and W_p contains no points F_P-\left\{f_p \right\}. For example, we can define W_p as in the above section “Bing’s Example G – Initial Discussion”.

Suppose that F_P is a G_\delta-set. Then F_P=\bigcap \limits_{i=1}^\infty O_i where each O_i is an open subset of F. Now for each p \in P, we have \left\{f_p \right\}=\bigcap \limits_{i=1}^\infty (O_i \cap W_p), contradicting the fact that the point f_p cannot be a G_\delta-point in the space F (and in the product space 2^Q). Thus F_P is not a G_\delta-set in the space F, leading to the conclusion that Bing’s Example G is not perfectly normal.

____________________________________________________________________

Bing’s Example G is not Metacompact

A space M is said to have caliber \omega_1 if for every uncountable collection \left\{U_\alpha: \alpha < \omega_1 \right\} of non-empty open subsets of M, there is an uncountable A \subset \omega_1 such that \bigcap \left\{U_\alpha: \alpha \in A \right\} \ne \varnothing. Any product of separable spaces has this property (see Topological Spaces with Caliber Omega 1). Thus the product space 2^Q has caliber \omega_1. Thus in the product space 2^Q, no collection of uncountably many non-empty open sets can be a point-finite collection (in fact cannot even be point-countable).

To see that the Example G is not metacompact, let \mathcal{W}=\left\{W_p: p \in P \right\} be a collection of open sets such that for p \in P, f_p \in W_p, W_p is open in the product topology of 2^Q and W_p contains no points F_P-\left\{f_p \right\}. For example, we can define W_p as in the above section “Bing’s Example G – Initial Discussion”.

Let W=\bigcup \mathcal{W}. Let \mathcal{V}=\mathcal{W} \cup \left\{\left\{ x \right\}: x \in F-W \right\}. Any open refinement of \mathcal{V} would contain uncountably many open sets in the product topology and thus cannot be point-finite. Thus the space F cannot be metacompact.

____________________________________________________________________

Reference

  1. Bing, R. H., Metrization of Topological Spaces, Canad. J. Math., 3, 175-186, 1951.
  2. Burke, D. K., A note on R. H. Bing’s example G, Top. Conf. VPI, Lectures Notes in Mathematics, 375, Springer Verlag, New York, 47-52, 1974.
  3. Engelking, R., General Topology, Revised and Completed edition, Heldermann Verlag, Berlin, 1989.
  4. Lewis, I. W., On covering properties of subspaces of R. H. Bing’s Example G, Gen. Topology Appl., 7, 109-122, 1977.
  5. Michael, E., Point-finite and locally finite coverings, Canad. J. Math., 7, 275-279, 1955.
  6. Willard, S., General Topology, Addison-Wesley Publishing Company, 1970.

____________________________________________________________________

\copyright \ \ 2012

Advertisements

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s