Bing’s Example G is the first example of a topological space that is normal but not collectionwise normal (see ). Example G was an influential example from an influential paper. The Example G and its subspaces had been extensively studied. In addition to being normal and not collectionwise normal, Example G is not perfectly normal and not metacompact. See the previous post “Bing’s Example G” for a basic discussion of Example G. In this post we focus on one subspace of Example G examined by Michael in . This subspace is normal, not collectionwise normal and not perfectly normal just like Example G. However it is metacompact. In , Michael proved that any metacompact collectionwise normal space is paracompact (metacompact was called pointwise paracompact in that paper). This subspace of Example G demonstrates that collectionwise normality in Michael’s theorem cannot be replaced by normality.
Bing’s Example G
For a more detailed discussion of Bing’s Example G in this blog, see the blog post “Bing’s Example G”. For the sake of completeness, we repeat the definition of Example G. Let be any uncountable set. Let be the set of all subsets of . Let be the set of all functions . Obviously is simply the Cartesian product of many copies of the two-point discrete space , i.e., . For each , define the function by the following:
, if and if
Let . Let be the set of all open subsets of in the product topology. The following is another topology on :
Bing’s Example G is the set with the topology . In other words, each is made an isolated point and points in retain the usual product open sets.
Michael’s Subspace of Example G
For each , let be the support of , i.e., . Michael in  considered the following subspace of .
Michael in  used the letter to denote the space . We choose another letter to distinguish it from Example G. The subspace consists of all points and all other such that for only finitely many . The space is normal and not collectionwise Hausdorff (hence not collectionwise normal and not paracompact). By eliminating points that have values of for infinitely many , we obtain a subspace that is metacompact. We discuss the following points:
- The space is normal.
- The space is not collectionwise Hausdorff and hence not collectionwise normal.
- The space is metacompact.
- The space is not perfectly normal.
The space is normal since the space that is Example G is hereditarily normal (see the section called Bing’s Example G is Completely Normal in the post “Bing’s Example G”).
To show that the space is not collectionwise Hausdorff, it is helpful to first look at as a subspace of the product space . The product space has the countable chain condition (CCC) since it is a product of separable spaces. Note that is dense in the product space . Thus as a subspace of the product space has the CCC.
In the space , the set is still a closed and discrete set. In the space , open sets containing points of are the same as product open sets in relative to the set . Since has CCC (as a subspace of the product space ), cannot have uncountably many pairwise disjoint open sets containing points of (in either the product topology or the Example G subspace topology). It follows that is not collectionwise Hausdorff. If it were, there would be uncountably many pairwise disjoint product open sets separating points in , which is not possible.
To see that is metacompact, let be an open cover of . For each , choose such that . For each , let . Let be the following:
Note that is a point-finite open refinement of . Each contains only one point of , namely . On the other hand, each with finite support can belong to at most finitely many .
The space is not perfectly normal. This point is alluded to in  by Michael and elsewhere in the literature, e.g. in Bing’s paper (see ) and in Engelking’s general topology text (see 5.53 on page 338 of ). In fact Michael indicated that one can obtain a perfectly normal example with the aforementioned properties using Example H defined in  instead of using the subspace defined here in this post.
- Bing, R. H., Metrization of Topological Spaces, Canad. J. Math., 3, 175-186, 1951.
- Engelking, R., General Topology, Revised and Completed edition, Heldermann Verlag, Berlin, 1989.
- Michael, E., Point-finite and locally finite coverings, Canad. J. Math., 7, 275-279, 1955.
- Willard, S., General Topology, Addison-Wesley Publishing Company, 1970.