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.
Completely Regular Spaces
The starting point is a completely regular space. A space is said to be completely regular if is a space and for each and for each closed subset of with , there is a continuous function such that and . Note that the axiom and the existence of the continuous function imply the axiom, which is equivalent to the property that single points are closed sets. Completely regular spaces are also called Tychonoff spaces.
Defining the Function Space
Let be a completely regular space. Let be the set of all real-valued continuous functions defined on the space . The set is naturally a subspace of the product space . Thus can be endowed with the subspace topology inherited from the product space . When this is the case, the function space is denoted by . The topology on is called the pointwise convergence topology.
Now we need a good handle on the open sets in the function space . A basic open set in the product space is of the form where each is an open subset of such that for all but finitely many (equivalently for only finitely many ). Thus a basic open set in is of the form:
where each is an open subset of and for all but finitely many . In addition, when , we can take to be an open interval of the form . To simplify notation, the basic open sets as described in (1) can also be notated by:
Thus when working with open sets in , we take to mean the set of all such that for each .
To make the basic open sets of more explicit, (1) or (1a) is translated as follows:
where is a finite set, for each , is an open interval of , and is the set of all such that .
There is another description of basic open sets that is useful. Let . Let be finite. Let . Let be defined as follows:
In proving results about , we can use basic open sets that are described in any one of the three forms (1), (2) and (3). If is a basic open subset of , as described in (1) or (1a), we use to denote the finite set of such that . The set is called the support of . The support for the basic open sets as described in (2) and (3) is already explicitly stated.
The theory of is a vast subject area. For a systematic introduction, see . One fundamental theme in function space theory is the study of how properties of and are related. The domain space and the function space are not on the same footing. The domain only has a topological structure. The function space carries a topology and two natural algebraic operations of addition and multiplication, making it a topological ring. In addition, 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 and , paying attention to the how the topological properties of and are related.
In addition to the pointwise convergence topology, there are other topologies that can be defined on , 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 inherited from the product space.
The space automatically inherits certain properties of the product space . Note that is dense in . The product has the countable chain condition (CCC) since it is a product of separable spaces. Hence always has the CCC, i.e., there are no uncountably many pairwise disjoint open subsets of , regardless what the domain space is. One consequence of the CCC is that is paracompact if and only if is Lindelof.
It is well known that is separable if and only if the cardinality of continuum. Since is dense in , is not separable if the cardinality of continuum. Thus is one way to get a CCC space that is not separable. There are non-separable where the cardinality of continuum. Obtaining such would require more than the properties of the product space ; using properties of would be necessary.
The properties of discussed so far are inherited from the product space. Refer to chapter one of  for other elementary properties of . See this post for a discussion of where is a separable metric space. See this post about a consequence of normality of .
- Arkhangelskii, A. V., Topological Function Spaces, Mathematics and Its Applications Series, Kluwer Academic Publishers, Dordrecht, 1992.