A compact space is said to be a Corson compact space if it can be embedded in a -product of real lines. All compact metric spaces are Corson. In this post, we present a non-metrizable example of a Corson compact space found in the literature, found in . A listing of other blog posts on Corson compact spaces is given at the end of this post.
For any infinite cardinal number , the -product of many copies of is the following subspace of :
A compact space is said to be a Corson compact space if it can be embedded in for some infinite cardinal . When , is simply , the product of countably many copies of the real lines. Any compact metrizable space can be embedded in ; see Theorem 4.2.10 in . Thus any compact metrizable space is Corson compact. One easily described non-metrizable Corson compact space is the one-point compactification of an uncountable discrete space.
Describing the example
We define a Corson compact space that is not metrizable. To define the example, let be a set of real numbers such that the cardinality of is . Let be a well ordering on the set such that for each , the initial segment is countable. In other words, the well-ordered set is of type . Let be the usual order on the real numbers. Consider the following collection of subsets of :
Note that if and only if this condition holds: if with , then
if and only if .
Thus by default, and for any . More importantly, every is a countable set. To see this, suppose that is uncountable. Then has a two-sided limit point, say (see this previous post). It does not matter whether . What matters is that for any open interval with , both and contain points of . Then there is a sequence of points of converging to and there is a sequence of points of converging to such that
Suppose that is a well-ordered set. Let such that is the least number that is an upper bound of . It must be the case that . If , then is an element of that has no immediate successor, a contradiction. If , then is not the least upper bound of . It follows that cannot be a well-ordered set. Then the orders and do not agree on the set . Thus no uncountable set can be in the family .
Consider the compact product space . For each , let be the support of the point , i.e., the set of all such that . Consider the following subspace of :
Note that where for all and that for each , where and for all . Since each is countable, is a subspace of the -product . Since is a subspace of the compact space , in order to show that is compact, we only need to show that is closed in . To this end, let . Then and do not coincide. Then there exist such that and cannot both be true. Suppose and . Consider the following open subset of :
It is clear that . Thus is a closed subspace of . Since is a compact space and is a subspace of a -product of real lines, is a Corson compact space.
If a compact space is metrizable then it is separable. Thus if we can show that is not separable, then is not metrizable. We show that no countable subspace of can be dense in . Let be a countable subspace of . Let , which is countable. Choose . Then the open set contains no point of . This completes the proof that is a non-metrizable Corson compact space.
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 (this post)
- Sigma-products of separable metric spaces are monolithic
- A short note on monolithic spaces
- Alster, K., Pol, R., On function spaces of compact subspaces of -products of the real line, Fund. Math., 107, 35-46 1980.
- Engelking, R., General Topology, Revised and Completed edition, Heldermann Verlag, Berlin, 1989.