The Euclidean spaces and are both Lindelof and separable. In fact these two properties are equivalent in the class of metrizable spaces. A space is metrizable if its topology can be induced by a metric. In a metrizable space, having one of these properties implies the other one. Any students in beginning topology courses who study basic notions such as the Lindelof property and separability must venture outside the confine of Euclidean spaces or metric spaces. The goal of this post is to present some elementary examples showing that these two notions are not equivalent.
All topological spaces under consideration are Hausdorff. Let be a space. Let . The set is said to be dense in if every nonempty open subset of contains some point of . The space is said to be separable if there is countable subset of that is also dense in . All Euclidean spaces are separable. For example, in the real line , every open interval contains a rational number. Thus the set of all rational numbers is dense in .
Let be a collection of subsets of the space . The collection is said to be a cover of if every point of is contained in some element of . The collection is said to be an open cover of if, in addition it being a cover, consists of open sets in .
Let be a cover of the space . Let . If the collection is also a cover of , we say that is a subcover of . The space is a Lindelof space (or has the Lindelof property) if every open cover of has a countable subcover.
The real is Lindelof. Both the Lindelof property and the separability of follows from the fact that the Euclidean topology on can be generated by a countable base (e.g. one countable base consists of all open intervals with rational endpoints). Now some non-Euclidean (and non-metrizable) examples.
Example 1 – A Lindelof space that is not separable
Let be any uncountable set. Let be a point that is not in , e.g., let . Define the space as follows. Let every point in be isolated, meaning any singleton set is declared open for any . An open neighborhood of the point is of the form where is a countable subset of .
It is clear that the resulting space is Lindelof since every open set containing contains all but countably many points of . It is also clear that no countable set can be dense in .
Even though this example is Lindelof, it is not hereditarily Lindelof since the subspace is uncountable discrete space.
In a previous post, we showed that the space defined in this example is a productively Lindelof space (meaning that its product with every Lindelof space is Lindelof).
A space is said to have the countable chain condition (CCC) if there are no uncountable family of pairwise disjoint open subsets. It is clear that any separable space has the CCC. It follows that the space in Example 1 does not have the CCC, since the singleton sets (with ) forms a pairwise disjoint collection of open sets, showing that the Lindelof property does not even imply the weaker property of having the CCC.
Example 2 – A separable space that is not Lindelof
The example here is the Tangent Disc Space (Niemytzki’s Tangent Disc Topology in ). The underlying set is the upper half plane (the x-axis and the plane above the x-axis). In other words, consider the following set:
Let and . The line is the x-axis and is the upper plane without the x-axis. We define a topology on such that as a subspace in this topology is Euclidean. The open neighborhoods of a point are of the form where is an open disc tangent to the x-axis at the point . The figure below illustrates how open neighborhoods at the x-axis are defined.
It is clear that the points with rational coordinates in the upper half plane form a dense set in the tangent disc topology. Thus is separable. In any Lindelof space, there are no uncountable closed and discrete subsets. Note that the x-axis is a closed and discrete subspace in the tangent disc space. Thus is not Lindelof.
Though separable, the Tangent Disc Space is not hereditarily separable since the x-axis is uncountable and discrete.
The Tangent Disc Space is an interesting example. For example, it is a completely regular space that is an example of a Moore space that is not normal. For these and other interesting facts about the Tangent Disc Space, see .
For the Lindelof property and the property of being separable, there are plenty of examples of spaces that possess only one of the properties. All three references indicated below are excellent places to look. The book by Steen and Seebach () is an excellent catalog of interesting spaces (many of them are elementary).
- Engelking, R., General Topology, Revised and Completed edition, 1989, Heldermann Verlag, Berlin.
- Steen, L. A., Seebach, J. A.,Counterexamples in Topology, 1995, Dover Edition, Dover Publications, New York.
- Willard, S., General Topology, 1970, Addison-Wesley Publishing Company.