We observe from the following statement two examples of Lindelof spaces that are not hereditarily Lindelof.
- Any product space contains a discrete subspace having the same cardinality as the number of factor spaces.
Using the above observation, by choosing the factor spaces judiciously, the product of uncountably many spaces is a handy way of obtaining Lindelof spaces (in some cases -compact spaces) that are not hereditarily Lindelof. For definition and basic information about product spaces, see this previous post.
All spaces under consideration are at least Hausdorff. For each , let be a space with at least two points. For each , fix two points . Then the product space contains a discrete subspace that has the same cardinality as the cardinality of the index set .
For each , define by the following:
Let . It follows that and that is a discrete space.
Whenever the index set is uncountable, the product space contains an uncountable discrete subspace. Thus even if the product space is Lindelof, one of its subspace cannot be Lindelof. Taking the product of uncountably many factor spaces is a handy way to obtain Lindelof space that is not hereditarily Lindelof. Some examples are shown below.
Let the index set be uncountable. To make the product space Lindelof, we can make every one of its factor compact. Thus the product space is compact and not hereditarily Lindelof.
Thus the product space , the product of many copies of the unit interval, is compact and not hereditarily Lindelof. Another example is , the product of many copies of
Another way to make the product space Lindelof is to make some of the factors compact such that the product of the remaining non-compact factors is Lindelof. Then the product space is essentially the product of a compact space and a Lindelof space, which is always Lindelof.
For example, let and let for all with . Then the product space is Lindelof since it is essentially the product of a compact space and a Lindelof space. However, the product is not hereditarily Lindelof.
In fact, the product space in the previous paragraph is -compact (i.e. the union of countably many compact sets). To make the example not -compact, simply make the first factor space a non-locally compact Lindelof space. For example, use the Sorgenfrey line or the space of the irrational numbers.