Elementary Examples of Lindelof Spaces and Separable Spaces

The Euclidean spaces \mathbb{R} and \mathbb{R}^n 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 X be a space. Let D \subset X. The set D is said to be dense in X if every nonempty open subset of X contains some point of D. The space X is said to be separable if there is countable subset of X that is also dense in X. All Euclidean spaces are separable. For example, in the real line \mathbb{R}, every open interval contains a rational number. Thus the set of all rational numbers \mathbb{Q} is dense in \mathbb{R}.

Let \mathcal{U} be a collection of subsets of the space X. The collection \mathcal{U} is said to be a cover of X if every point of X is contained in some element of \mathcal{U}. The collection \mathcal{U} is said to be an open cover of X if, in addition it being a cover, \mathcal{U} consists of open sets in X.

Let \mathcal{U} be a cover of the space X. Let \mathcal{V} \subset \mathcal{U}. If the collection \mathcal{V} is also a cover of X, we say that \mathcal{V} is a subcover of \mathcal{U}. The space X is a Lindelof space (or has the Lindelof property) if every open cover of X has a countable subcover.

The real \mathbb{R} is Lindelof. Both the Lindelof property and the separability of \mathbb{R} follows from the fact that the Euclidean topology on \mathbb{R} 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.

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Example 1 – A Lindelof space that is not separable
Let X be any uncountable set. Let p be a point that is not in X, e.g., let p=\left\{ X \right\}. Define the space Y = \left\{p\right\} \cup X as follows. Let every point in X be isolated, meaning any singleton set \left\{ x \right\} is declared open for any x \in X. An open neighborhood of the point p is of the form \left\{p\right\} \cup W where X-W is a countable subset of X.

It is clear that the resulting space Y is Lindelof since every open set containing p contains all but countably many points of X. It is also clear that no countable set can be dense in Y.

Even though this example Y is Lindelof, it is not hereditarily Lindelof since the subspace X is uncountable discrete space.

In a previous post, we showed that the space Y defined in this example is a productively Lindelof space (meaning that its product with every Lindelof space is Lindelof).

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Remark

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 Y in Example 1 does not have the CCC, since the singleton sets \left\{ x \right\} (with x \in X) forms a pairwise disjoint collection of open sets, showing that the Lindelof property does not even imply the weaker property of having the CCC.

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Example 2 – A separable space that is not Lindelof
The example here is the Tangent Disc Space (Niemytzki’s Tangent Disc Topology in [2]). 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:

\displaystyle . \ \ \ \ \ X=\left\{(x,y) \in \mathbb{R}^2: y \ge 0 \right\}

Let \displaystyle X_u=\left\{(x,y) \in \mathbb{R}^2: y>0 \right\} and T=\left\{(x,0): x \in \mathbb{R} \right\}. The line T is the x-axis and X_u is the upper plane without the x-axis. We define a topology on X such that X_u as a subspace in this topology is Euclidean. The open neighborhoods of a point p=(x,0) \in T are of the form \left\{p \right\} \cup D where D is an open disc tangent to the x-axis at the point p. 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 X_u form a dense set in the tangent disc topology. Thus X is separable. In any Lindelof space, there are no uncountable closed and discrete subsets. Note that the x-axis T is a closed and discrete subspace in the tangent disc space. Thus X is not Lindelof.

Though separable, the Tangent Disc Space is not hereditarily separable since the x-axis T 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 [2].

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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 ([2]) is an excellent catalog of interesting spaces (many of them are elementary).
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Reference

  1. Engelking, R., General Topology, Revised and Completed edition, 1989, Heldermann Verlag, Berlin.
  2. Steen, L. A., Seebach, J. A.,Counterexamples in Topology, 1995, Dover Edition, Dover Publications, New York.
  3. Willard, S., General Topology, 1970, Addison-Wesley Publishing Company.
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A proof about the Michael Line

The original post about the Michael line is now replaced by the following newer posts. You can go to these newer posts using the following links.

“Michael Line Basics”

“Finite and Countable Products of the Michael Line”

“Bernstein Sets and the Michael Line”

“The Michael Line and the Continuum Hypothesis”