Working with the function space Cp(X)

This post provides basic information about the space of real-valued continuous functions with the pointwise convergence topology. The goal is to discuss the setting and to define the standard basic open sets in the function space, providing background information for subsequent posts.

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Completely Regular Spaces

The starting point is a completely regular space. 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.

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Defining the Function Space $C_p(X)$

Let $X$ be a completely regular 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 $\prod_{x \in X} \mathbb{R}=\mathbb{R}^X$. Thus $C(X)$ can be endowed with the subspace topology inherited from the product space $\mathbb{R}^X$. When this is the case, the function space $C(X)$ is denoted by $C_p(X)$. The topology on $C_p(X)$ is called the pointwise convergence topology.

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 $\prod_{x \in X} U_x$ where each $U_x$ is an open subset of $\mathbb{R}$ such that $U_x = \mathbb{R}$ 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:

$C(X) \cap \biggl(\prod_{x \in X} U_x \biggr) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)$

where each $U_x$ is an open subset of $\mathbb{R}$ 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 simplify notation, the basic open sets as described in (1) can also be notated by:

$\prod_{x \in X} U_x \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1a)$

Thus when working with open sets in $C_p(X)$, we take $\prod_{x \in X} U_x$ to mean the set of all $f \in C(X)$ such that $f(x) \in U_x$ for each $x \in X$.

To make the basic open sets of $C_p(X)$ more explicit, (1) or (1a) is translated as follows:

$\bigcap_{x \in F} \ [x, O_x] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2)$

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 $f(x) \in O_x$.

There is another description of basic open sets that is useful. Let $f \in C_p(X)$. Let $F \subset X$ be finite. Let $\epsilon>0$. Let $B(f,F,\epsilon)$ be defined as follows:

$B(f,F,\epsilon)=\left\{g \in C(X): \forall \ x \in F, \lvert f(x)-g(x) \lvert< \epsilon \right\} \ \ \ \ \ \ \ (3)$

In proving results about $C_p(X)$, we can use basic open sets that are described in any one of the three forms (1), (2) and (3). If $U$ is a basic open subset of $C_p(X)$, as described in (1) or (1a), we use $supp(U)$ to denote the finite set of $x \in X$ such that $U_x \ne \mathbb{R}$. The set $supp(U)$ is called the support of $U$. The support for the basic open sets as described in (2) and (3) is already explicitly stated.

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Basic Discussion

The theory of $C_p(X)$ is a vast subject area. For a systematic introduction, see [1]. One fundamental theme in function space theory is the study of how properties of $X$ and $C_p(X)$ are related. The domain space $X$ and the function space $C(X)$ are not on the same footing. The domain $X$ only has a topological structure. The function space $C_p(X)$ carries a topology and two natural algebraic operations of addition and multiplication, making it a topological ring. In addition, $C_p(X)$ can be regarded as a topological group, or a linear topological space. In this post and in many subsequent posts, we narrow the focus to the topological properties of $X$ and $C_p(X)$, paying attention to the how the topological properties of $X$ and $C_p(X)$ are related.

In addition to the pointwise convergence topology, there are other topologies that can be defined on $C(X)$, e.g., the compact-open topology, the topology of uniform convergence and others. Both the pointwise convergence topology and the compact-open topology are examples of set-open topologies. In this post and in many of the subsequent posts, the focus is on the pointwise convergence topology, i.e., the subspace topology on $C(X)$ inherited from the product space.

The space $C_p(X)$ automatically inherits certain properties of the product space $\mathbb{R}^X$. Note that $C(X)$ is dense in $\mathbb{R}^X$. The product $\mathbb{R}^X$ has the countable chain condition (CCC) since it is a product of separable spaces. Hence $C_p(X)$ always has the CCC, i.e., there are no uncountably many pairwise disjoint open subsets of $C_p(X)$, regardless what the domain space $X$ is. One consequence of the CCC is that $C_p(X)$ is paracompact if and only if $C_p(X)$ is Lindelof.

It is well known that $\mathbb{R}^X$ is separable if and only if the cardinality of $X$ $\le$ continuum. Since $C(X)$ is dense in $\mathbb{R}^X$, $C_p(X)$ is not separable if the cardinality of $X$ $>$ continuum. Thus $C_p(X)$ is one way to get a CCC space that is not separable. There are non-separable $C_p(X)$ where the cardinality of $X$ $\le$ continuum. Obtaining such $C_p(X)$ would require more than the properties of the product space $\mathbb{R}^X$; using properties of $X$ would be necessary.

The properties of $C_p(X)$ discussed so far are inherited from the product space. Refer to chapter one of [1] for other elementary properties of $C_p(X)$. See this post for a discussion of $C_p(X)$ where $X$ is a separable metric space. See this post about a consequence of normality of $C_p(X)$.

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Reference

1. Arkhangelskii, A. V., Topological Function Spaces, Mathematics and Its Applications Series, Kluwer Academic Publishers, Dordrecht, 1992.

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$\copyright \ 2014 \text{ by Dan Ma}$