In this post, we discuss an example of a function space that is normal and not Lindelof (as indicated in the title). Interestingly, much more can be said about this function space. In this post, we show that there exists a space such that
- is collectionwise normal and not paracompact,
- is not Lindelof but contains a dense Lindelof subspace,
- is not first countable but is a Frechet space,
- As a corollary of the previous point, cannot contain a copy of the compact space ,
- is homeomorphic to ,
- is not hereditarily normal,
- is not metacompact.
A short and quick description of the space is that is the one-point Lindelofication of an uncountable discrete space. As shown below, the function space is intimately related to a -product of copies of real lines. The results listed above are merely an introduction to this wonderful example and are derived by examining the -products of copies of real lines. Deep results about -product of real lines abound in the literature. The references listed at the end are a small sample. Example 3.2 in  is another interesting illustration of this example.
We now define the domain space . In the discussion that follows, the Greek letter is always an uncountable cardinal number. Let be a set with cardinality . Let be a point not in . Let . Consider the following topology on :
- Each point in an isolated point, and
- open neighborhoods at the point are of the form where is countable.
It is clear that is a Lindelof space. The Lindelof space is sometimes called the one-point Lindelofication of the discrete space since it is a Lindelof space that is obtained by adding one point to a discrete space.
Consider the function space . See this post for general information on the pointwise convergence topology of for any completely regular space .
All the facts about mentioned at the beginning follow from the fact that is homeomorphic to the -product of many copies of the real lines. Specifically, is homeomorphic to the following subspace of the product space .
Thus understanding the function space is a matter of understanding a -product of copies of the real lines. First, we establish the homeomorphism and then discuss the properties of indicated above.
For each , it is easily seen that there is a countable set such that for all . Let . Then each has non-zero values only on a countable subset of . Naturally, and are homeomorphic.
We claim that is homeomorphic to . For each , define . Here, is the function such that for all . Clearly is well-defined and . It can be readily verified that is a one-to-one map from onto . It is not difficult to verify that both and are continuous.
We use the notation to mean that the spaces and are homeomorphic. Then we have:
Thus . This completes the proof that is topologically the -product of many copies of the real lines.
Looking at the -Product
Understanding the function space is now reduced to the problem of understanding a -product of copies of the real lines. Most of the facts about -products that we need have already been proved in previous blog posts.
In this previous post, it is established that the -product of separable metric spaces is collectionwise normal. Thus is collectionwise normal. The -product of spaces, each of which has at least two points, always contains a closed copy of with the ordered topology (see the lemma in this previous post). Thus contains a closed copy of and hence can never be paracompact (and thus not Lindelof).
Consider the following subspace of the -product :
In this previous post, it is shown that is a Lindelof space. Though is not Lindelof, it has a dense Lindelof subspace, namely .
A space is first countable if there exists a countable local base at each point . A space is a Frechet space (or is Frechet-Urysohn) if for each , if where , then there exists a sequence of points of such that the sequence converges to . Clearly, any first countable space is a Frechet space. The converse is not true (see Example 1 in this previous post).
For any uncountable cardinal number , the product is not first countable. In fact, any dense subspace of is not first countable. In particular, the -product is not first countable. In this previous post, it is shown that the -product of first countable spaces is a Frechet space. Thus is a Frechet space.
As a corollary of the previous point, cannot contain a homeomorphic copy of any space that is not Frechet. In particular, it cannot contain a copy of any compact space that is not Frechet. For example, the compact space is not embeddable in . The interest in compact subspaces of is that any compact space that is topologically embeddable in a -product of real lines is said to be Corson compact. Thus any Corson compact space is a Frechet space.
It can be readily verified that
Thus . In particular, due to the following observation:
As a result of the peculiar fact that , it can be concluded that , though normal, is not hereditarily normal. This follows from an application of Katetov’s theorem. The theorem states that if is hereditarily normal, then either is perfectly normal or every countably infinite subset of is closed and discrete (see this previous post). The function space is not perfectly normal since it contains a closed copy of . On the other hand, there are plenty of countably infinite subsets of that are not closed and discrete. As a Frechet space, has many convergent sequences. Each such sequence without the limit is a countably infinite set that is not closed and discrete. As an example, let be an infinite subset of and consider the following:
where is such that and for each with . Note that is not closed and not discrete since the points in converge to where is the zero-function. Thus is not hereditarily normal.
It is well known that collectionwise normal metacompact space is paracompact (see Theorem 5.3.3 in  where metacompact is referred to as weakly paracompact). Since is collectionwise normal and not paracompact, can never be metacompact.
- Arkhangelskii, A. V., Topological Function Spaces, Mathematics and Its Applications Series, Kluwer Academic Publishers, Dordrecht, 1992.
- Bella, A., Masami, S., Tight points of Pixley-Roy hyperspaces, Topology Appl., 160, 2061-2068, 2013.
- Corson, H. H., Normality in subsets of product spaces, Amer. J. Math., 81, 785-796, 1959.
- Engelking, R., General Topology, Revised and Completed edition, Heldermann Verlag, Berlin, 1989.