An observation about hereditarily separable function spaces

For any completely regular space X, by C_p(X) we mean the space of all real-valued continuous functions on X endowed with the pointwise convergent topology. It is known that C_p(X) is hereditarily separable if and only if X^n is hereditarily Lindelof for all positive integer n if and only if X^\omega is hereditarily Lindelof where \omega is the first infinite ordinal (a result that follows from a theorem of Zenor in [4]). This result points to a duality between hereditary separability of the function space C_p(X) and the hereditary Lindelof property of the domain space X and is restated below.

Theorem
Let X be a completely regular space. Then the following conditions are equivalent:

  1. C_p(X) is hereditarily separable.
  2. X^n is hereditarily Lindelof for all positive integer n.
  3. X^\omega is hereditarily Lindelof.

As an introduction to this theorem, we present the proof to one direction of this theorem for n=1.

Observation
Let X be any completely regular space. We have the following obervation:

If C_p(X) is hereditarily separable, then X is hereditarily Lindelof.

Suppose X is not hereditarily Lindelof. We aim to show that C_p(X) is not hereditarily separable by producing a non-separable subspace \mathcal{F} of C_p(X).

Let Y \subset X be a subspace that is not Lindelof. Let \mathcal{U} be a collection of open subsets of X such that \mathcal{U} covers Y and no countable subcollection of \mathcal{U} covers Y.

For each y \in Y, choose U_y \in \mathcal{U} such that y \in U_y. By the completely regularity of X, choose a continuous f_y: X \rightarrow \mathbb{R} such that f_y maps X-U_y to 0 and f_y(y)=1. Let \mathcal{F}=\left\{f_y:y \in X \right\}. It can be shown that \mathcal{F} is a non-separable subspace of C_p(X). That is, no countable subset of \mathcal{F} can be dense in \mathcal{F}. \blacksquare

For any completely regular space X, it is also known (see [2]) that C_p(X) is separable if and only if X has a weaker separable metrizable topology (i.e. X has a weaker topology such that X with this weaker topology is a separable metrizable space). The result in [2] combined with the observation presented here provides a way to obtain sepearable C_p(X) that is not hereditarily separable. Look for any X that is not hereditarily Lindelof but has a weaker separable metrizable topology. One such example is the Michael Line.

The observation we make here is a rather weak result. The double arrow space Z is hereditarily Lindelof. Yet C_p(Z) is not even separable since Z is compact space that is not metrizable. Note that Z^2 is not hereditarily Lindelof since it contains a copy of the Sorgenfrey plane (see the previous post on double arrow space).

Reference

  1. Engelking, R., General Topology, Revised and Completed edition, 1989, Heldermann Verlag, Berlin.
  2. Noble, N., The density character of function spaces, Proc. Amer. Math. Soc. 42:1 (1974) 228-233.
  3. Willard, S., General Topology, 1970, Addison-Wesley Publishing Company.
  4. Zenor, P., Hereditarily m-separability and the hereditarily m-lindelof property in product spaces and function spaces, Fund. Math. 106 (1980), 175-180
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