A space is said to have countable chain condition (ccc) if every disjoint collection of open subsets of is countable. For convenience, we refer to spaces that have countable chain condition as ccc spaces. It is easy to verify that separable spaces are ccc spaces. We present a specific way of generating spaces that always have ccc but are not separable. These spaces are the sigma products of separable spaces.
The product of separable spaces always have ccc (see Product of Spaces with Countable Chain Condition). However, the product of separable spaces is not separable when the number of factors is greater than continuum. Thus one way to get an example of ccc but not separable space is to take the product of more than continuum many separable spaces. For example, if is the cardinality of continuum, , the product of many copies of , is a space that has ccc but is not separable.
Another example is obtained from taking sigma product of separable spaces. Let be a collection of spaces where is some index set. Consider the product space . Fix a point in the product. The sigma-product about the point is denoted by and is the following subspace of the product space :
To obtain the desired example, let be an uncountable index set and let each be separable. The product space has ccc. Note that is always dense in the product space . Thus the sigma-product has ccc since it is a dense subspace of a ccc space. On the other hand, is never separable as long as there are uncountably many spaces .
As specific example, take for each and let the fixed point be such that for all . The resulting is a ccc space that is not separable. Of course, in this case is the set of all such that for at most countably many .
One interesting note about the sigma-product is that the overall product space is an example of a separable but not hereditarily separable space. Another interesting point is that is a countably compact non-compact space (see A note about sigma-product of compact spaces).
Previous discussion of CCC spaces in this blog:
Product of Spaces with Countable Chain Condition
- Engelking, R., General Topology, Revised and Completed edition, Heldermann Verlag, Berlin, 1989.
- Willard, S., General Topology, Addison-Wesley Publishing Company, 1970.