Completely Regular Spaces and Function Spaces

This is a continuation of a discussion on completely regular spaces (continuing from this previous post). These spaces are an integral part of many discussions involving topological spaces and/or properties. Some notions are contingent on the existence of certain real-valued continuous functions. Discussion of such notions can be greatly facilitated by working in the class of completely regular spaces. One example given in a previous post is on a discussion of pseudocompact spaces. Another example for requiring complete regularity is when working with function spaces. When the object being studied is the space of real-valued continuous functions defined on a topological space X, it is desirable to have enough continuous functions in the function space being studied. In this post, we illustrate this point by giving the proofs of two simple results in C_p(X), the space of real-valued continuous functions endowed with the pointwise convergence topology.

Basic references are [2] and [4]. Refer to [3] for a discussion of where complete regularity is placed among the separation axioms. An in-depth treatment for C_p theory is [1].
________________________________________________________________________

Completely Regular Spaces

A space X is said to be completely regular if X is a T_0 space and for each x \in X and for each closed subset A of X with x \notin A, there is a continuous function f:X \rightarrow [0,1] such that f(A) \subset \left\{0 \right\} and f(x)=1. Note that the T_0 axiom and the existence of the continuous function imply the T_1 axiom, which is equivalent to the property that single points are closed sets. Completely regular spaces are also called Tychonoff spaces.

________________________________________________________________________

Function Spaces

Let X be a space. Let C(X) be the set of all real-valued continuous functions defined on the space X. The set C(X) is naturally a subspace of the product space \prod \limits_{x \in X} Y_x where each Y_x = \mathbb{R}. We can also write \prod \limits_{x \in X} Y_x = \mathbb{R}^X. Thus C(X) can be endowed with the subspace topology inherited from the product space \mathbb{R}^X. When this is the case, the resulting function space is denoted by C_p(X).

Now we need a good handle on the open sets in the function space C_p(X). A basic open set in the product space \mathbb{R}^X is of the form \prod \limits_{x \in X} U_x where each U_x is open in \mathbb{R} and U_x = \mathbb{R} for all but finitely many x \in X (equivalently U_x \ne \mathbb{R} for only finitely many x \in X). Thus a basic open set in C_p(X) is of the form:

(1) \ \ \ \ \ \ \ \ C(X) \cap \prod \limits_{x \in X} U_x

where each U_x is open in \mathbb{R} and U_x = \mathbb{R} for all but finitely many x \in X. In addition, when U_x \ne \mathbb{R}, we can take U_x to be an open interval of the form (a,b). To make the basic open sets of C_p(X) more explicit, (1) is translated as follows:

(2) \ \ \ \ \ \ \ \ \bigcap \limits_{x \in F} [x, O_x]

where F \subset X is a finite set, for each x \in F, O_x is an open interval of \mathbb{R}, and [x,O_x] is the set of all f \in C(X) such that f(x) \in O_x. In proving results about C_p(X), we can use basic open sets that are described in (2).

________________________________________________________________________

Completely Regular Spaces and Function Spaces

We now give two simple results in C_p(X) that are good illustrations that complete regularity is the ideal setting for the domain space of C_p function spaces. In Theorem 1, complete regularity is used to generate a non-separable subspace of the function space given a non-Lindelof subspace of the domain space. In Theorem 2, complete regularity is used to generate a non-Lindelof subspace of the function space given a non-separable subspace of the domain space.

Theorem 1
Let X be a completely regular space. Then if C_p(X) is hereditarily separable, then X is hereditarily Lindelof.

Proof
We show that if X has a subspace that is not Lindelof, then there is a subspace of C_p(X) that is not separable. We use complete regularity to generate the continuous functions that form a non-separable subspace.

Let Y \subset X be a non-Lindelof subspace. There exists an open cover \mathcal{U} of Y such that \mathcal{U} has no countable subcover. Open sets in \mathcal{U} are open sets in Y. We wish to expand these to open sets in X. Let \mathcal{U}^* be the collection of open subsets U of X such that U \cap Y \in \mathcal{U}. It is clear that no countable subcollection of \mathcal{U}^* can cover Y.

For each y \in Y, choose U(y) \in \mathcal{U}^* such that y \in U(y). Here’s where we use complete regularity. For each y \in Y, there is a continuous function f_y:X \rightarrow [0,1] such that f_y(X-U(y)) \subset \left\{0 \right\} and f_y(y) =1. Let T=\left\{f_y:y \in Y \right\}, which is a subspace of C_p(X).

We claim that T is not separable. To see this, let A=\left\{g_1,g_2,g_3,\cdots \right\} be a countable subset of T such that for each i, g_i is obtained from the point y(i) \in Y and the open set U_i=U(y(i)), i.e., g_i=f_{y(i)}. Note that \left\{U_1,U_2,U_3,\cdots \right\} cannot be a cover of Y. Let a \in Y be a point that is not in all U_i.

Consider the function f_a chosen above using complete regularity. Note that f_a(a)=1 and f_a(X-U(a)) \subset \left\{0 \right\}. On the other hand, g_i(a)=f_{y(i)}(a)=0 for all i since a \notin U_i for all i. This means that f_a is not in the closure of A. For example, [a,(0.9,1.1)] is a basic open set containing f_a such that g_i \notin [a,(0.9,1.1)] for all i. Thus no countable subset of T can be dense in T, completing the proof. \blacksquare

Theorem 2
Let X be a completely regular space. Then if C_p(X) is hereditarily Lindelof, then X is hereditarily separable.

Proof
Let Y be a non-separable subspace of X. For each countable A \subset Y, there must be some point y(A) \in Y such that y(A) is not a member of the closure of A (relative to Y). For each countable A \subset Y, let \overline{A} be the closure of A in the entire space X. Clearly y(A) \notin \overline{A}.

Now apply the complete regularity of the space X. For each countable A \subset Y, let f_A: X \rightarrow [0,1] be continuous such that f_A(\overline{A}) \subset \left\{0 \right\} and f_A(y(A))=1. Let W be the set of all f_A where A \subset Y is countable.

We now show that W is a non-Lindelof subspace of C_p(X). For each f_A \in W, let U_A=[y(A),(0.9, 1.1)] \cap W, which is open in C_p(X) and contains f_A. Let \mathcal{U} be the collection of all such open sets U_A.

Then \mathcal{U} is an open cover of W that has no countable subcover. To see this, suppose we have \left\{U_1,U_2,U_3,\cdots \right\} such that for each i, U_i=U_{A(i)} where A(i) \subset Y is countable. Then let B be the set of all A(i) and all points y(A(i)). Note that the set B is still a countable subset of Y. Consider the point y(B) and the continuous function f_B, which is a member of W. We have f_B(y(B))=1 and f_B(t)=0 for all t \in \overline{B}. Note that each y(A(i)) \in B and thus f_B(y(A(i))=0 for all i. This shows that f_B \notin U_i=U_{A(i)} for all i. Then \mathcal{U} is an open cover of W that has no countable subcover, leading to the conclusion that W is a non-Lindelof subspace of C_p(X). \blacksquare

________________________________________________________________________

Remark

The above proofs of Theorem 1 and Theorem 2 are only meant as demonstration of the role played by complete regularity in working with function spaces. They are much weaker versions of a deeper result. These two results can serve to motivate a deeper result that explores the relationship between the hereditary separability (respectively hereditary Lindelof property) of the domain space and the hereditary Lindelof property (respectively the hereditary separability) of the function space. The following theorem is the countable version of a theorem found in [5].

Theorem 3
Let X be a completely regular space. The following conditions are equivalent.

  1. C_p(X) is hereditarily separable (respectively hereditarily Lindelof).
  2. X^\omega is hereditarily Lindelof (respectively hereditarily separable).
  3. For each positive integer n, X^n is hereditarily Lindelof (respectively hereditarily separable).

________________________________________________________________________

Reference

  1. Arhangel’skii, A., Topological Function Spaces, Kluwer Academic Publishers, Boston, 1992.
  2. Engelking, R., General Topology, Revised and Completed edition, Heldermann Verlag, Berlin, 1989.
  3. Steen, L. A., Seebach, J. A., Counterexamples in Topology, Dover Publications, Inc., New York, 1995.
  4. Willard, S., General Topology, Addison-Wesley Publishing Company, 1970.
  5. Zenor, P., Hereditarily m-separability and the hereditarily m-lindelof property in product spaces and function spaces, Fund. Math. 106, 175-180, 1980.
Advertisements

One thought on “Completely Regular Spaces and Function Spaces

  1. Pingback: Normality in Cp(X) | Dan Ma's Topology Blog

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s