Let be the -product of many copies of the real lines where is any infinite cardinal number. Any compact space that can be embedded in for some is said to be a Corson compact space. Corson compact spaces play an important role in functional analysis. Corson compact spaces are also very interesting from a topological point of view. Some of the properties of Corson compact spaces are inherited (as subspaces) from the -product . One such property is the property that the -product is monolithic, which implies that the closure of any countable subspace of is metrizable.
Previous blog posts on -products:
- Sigma-product of separable metric spaces is collectionwise normal
- Sigma-products of first countable spaces
A previous blog post on monolithic spaces: A short note on monolithic spaces. A listing of other blog posts on Corson compact spaces is given at the end of this post.
Let be an infinite cardinal number. For each , let be a topological space. Let . The -product of the spaces about the base point is defined as follows:
If each and if the base point is such that for all , then we use the notation for , i.e., is defined as follows:
A compact space is said to be a Corson compact space if it can be embedded in the -product for some infinite cardinal .
A space is monolithic if for every subspace of , the density of equals the network weight of , i.e., . A space is strongly monolithic if for every subspace of , the density of equals the weight of , i.e., . See the previous post called A short note on monolithic spaces.
The proof of the fact that -product of separable metrizable spaces is monolithic can be worked out quite easily from definitions. Interested readers are invited to walk through the proof. For the sake of completeness, we prove the following theorem.
Suppose that for each , is a separable metric space. Then the -product is strongly monolithic.
Proof of Theorem 1
Let be the base point of the -product . For each , let be the support of the point , i.e., the set of all such that . Let Y be a subspace of . We show that .
Let be a dense subspace of such that . Note that (closure is taken in ). Let . Clearly . Consider the following subspace of :
It is clear that is a closed subspace of . Since , the closure of (closure in or in ) is a subspace of . Thus . Note that . Since each has a countable base, the product space has a base of cardinality . Thus has weight . Since , both and have weights . We have . Note that always holds. Therefore .
For any infinite cardinal , the -product is strongly monolithic.
Any Corson compact space is strongly monolithic.
Blog posts on Corson compact spaces
- Basic topological properties of Corson compact spaces
- Every Corson compact space has a dense first countable subspace
- An example of a non-metrizable Corson compact space
- Sigma-products of separable metric spaces are monolithic (this post)
- A short note on monolithic spaces