Let be an uncountable cardinal. Let be the Cartesian product of many copies of the real line. This product space is not normal since it contains as a closed subspace. However, there are dense subspaces of are normal. For example, the -product of copies of the real line is normal, i.e., the subspace of consisting of points which have at most countably many non-zero coordinates (see this post). In this post, we look for more normal spaces among the subspaces of that are function spaces. In particular, we look at spaces of continuous real-valued functions defined on a separable metrizable space, i.e., the function space where is a separable metrizable space.
For definitions of basic open sets and other background information on the function space , see this previous post.
when is a separable metric space
In the remainder of the post, denotes a separable metrizable space. Then, is more than normal. The function space has the following properties:
- Lindelof (hence paracompact and collectionwise normal),
- hereditarily Lindelof (hence hereditarily normal),
- hereditarily separable,
- perfectly normal.
All such properties stem from the fact that has a countable network whenever is a separable metrizable space.
Let be a topological space. A collection of subsets of is said to be a network for if for each and for each open with , there exists some such that . A countable network is a network that has only countably many elements. The property of having a countable network is a very strong property, e.g., having all the properties listed above. For a basic discussion of this property, see this previous post and this previous post.
To define a countable network for , let be a countable base for the domain space . For each and for any open interval in the real line with rational endpoints, consider the following set:
There are only countably many sets of the form . Let be the collection of sets, each of which is the intersection of finitely many sets of the form . Then is a network for the function space . To see this, let where is a basic open set in where is finite and each is an open interval with rational endpoints. For each point , choose with such that . Clearly . It follows that .
Examples include , and . All three can be considered subspaces of the product space where is the cardinality of the continuum. This is true for any separable metrizable . Note that any separable metrizable can be embedded in the product space . The product space has cardinality . Thus the cardinality of any separable metrizable space is at most continuum. So is the subspace of a product space of continuum many copies of the real lines, hence can be regarded as a subspace of .
A space has countable extent if every closed and discrete subset of is countable. The -product of the separable metric spaces is a dense and normal subspace of the product space . The normal space has countable extent (hence collectionwise normal). The examples of discussed here are Lindelof and hence have countable extent. Many, though not all, dense normal subspaces of products of separable metric spaces have countable extent. For a dense normal subspace of a product of separable metric spaces, one interesting problem is to find out whether it has countable extent.