Let be a product space. Let be a continuous function where is a topological space. In this post, we discuss what it means for the continuous function to depend on countably many coordinates and then discuss some conditions that we can impose on the product space and on the range space to ensure that every continuous defined on the product space will depend on countably many coordinates. This notion of a continuous function depending on countably many coordinates is equivalent to factoring the continuous function into the composition of a projection map and a continuous function defined on a countable subproduct (see Lemma 1 below).
Let’s set some notation about the product space we work with in this post. Let be a product space. Let be a topological space. Let be continuous. For any , is the natural projection from the full product space into the subproduct . Standard basic open sets of are of the form where each is open in and that for all but finitely many . We use to denote the finite set of where .
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Factoring a Continuous Map
The function is said to depend on countably many coordinates if there exists a countable set such that for any , if for all , then . Suppose is instead defined on a subspace of . The function is said to depend on countably many coordinates if there exists a countable such that for any , if for all , then .
We have the following lemmas.
Lemma 1

Let be a product space. Let be a topological space. Let be continuous. Then the following are equivalent.
 There exists a countable such that for any , if for all , then .
 There exists a countable such that where is continuous.
Lemma 1a

Let be a product space. Let be a topological space. Let be a dense subspace of . Let be continuous. Then the following are equivalent.
 There exists a countable such that for any , if for all , then .
 There exists a countable such that where is continuous.
It is straightforward to verify Lemma 1 and Lemma 1a. We use condition 1 to define what it means for a function to be dependent on countably many coordinates. Both lemmas indicate that either condition is a valid definition. These two lemmas also indicate why the notion being discussed can be called a factorization notion.
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When a Continuous Map Can Be Factored
We discuss some conditions that we can place on the product space and on the range space so that any continuous map depends on countably many coordinates. We prove the following theorem.
Theorem 1

Let be a product space such that each factor is a separable space. Let be a second countable space (i.e. having a countable base). Then for any dense subspace of , any continuous function depends on countably many coordinates, i.e., either one of the conditions in Lemma 1a holds.
Before stating the main theorem, we need one more lemma. Let . The set is said to depend on countably many coordinates if there exists a countable such that for any and for any , if for all , then .
When we try to determine whether a function , where , can be factored, we will need to decide whether a set depends on countably many coordinates. Let . The set is said to depend on countably many coordinates if there exists a countable such that for any and for any , if for all , then . We have the following lemma.
Lemma 2

Let be a product space with the countable chain condition. Let be a dense subspace of .
 Let be an open subset of . Then depends on countably many coordinates.
 Let be an open subset of . Then depends on countably many coordinates (closure in ).
Proof of Lemma 2
Proof of Part 1
Let be open. Let be a collection of pairwise disjoint open subsets of the open set such that is maximal with this property, i.e., if you throw one more open set into , it will be no longer pairwise disjoint. Let . Since is maximal, . Since has the countable chain condition, is countable.
Let . The set is a countable subset of since is the union of countably many finite sets. We have the following claims.
Claim 1
The open set depends on the coordinates in .
Let and such that for all . We need to show that . Firstly, for some . It follows that for all . Thus . This completes the proof of Claim 1.
Claim 2
The set depends on the coordinates in .
Let and such that for all . We need to show . To this end, let be a standard basic open set with . The goal is to find some . Define such that for all and for all . Then . Since , there exists some . Define such that for all and for all . Since , . On the other hand, . This completes the proof of Claim 2.
As noted above, . Thus depends on countably many coordinates, namely the coordinates in the set . This completes the proof of Part 1.
Proof of Part 2
For any , let denote the closure of in . Let denote the closure of in . Let be open. Let be open in such that . By Part 1, depends on countably many coordinates, say the coordinates in the countable set . This means that for any and for any , if for all , then . Thus for any and for any , if for all , then . If we have , then we are done. So we only need to show that if and , then . This is why we need to assume is dense in .
Let and . Let be an open subset of with . There exists an open subset of such that . Then . Note that is open and . Since is dense in , must contain points of . These points of are also points of . Thus contains points of . It follows that . This concludes the proof of Part 2.
Proof of Theorem 1
Let be a dense subspace of . Let be continuous. Let be a countable base for the separable metrizable space . By Lemma 2 Part 2, for each , depends on countably many coordinates, say the countable set . Let .
We claim that is a countable set of coordinates we need. Let such that for all . We need to show that . Suppose . Choose such that
 for each
This is possible since is a second countable space. Then for some . Furthermore, . Since is continuous, . Therefore, . On the other hand, depends on the countably many coordinates in . We assume above that for all . Thus for all . This means that , a contradiction. It must be that case that .
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Another Version
We state another version of Theorem 1 that will be useful in some situations.
Theorem 2

Let be a product space such that each factor is a separable space. Let be a second countable space. Let be a dense subspace of . Let be any continuous function. Then the function depends on countably many coordinates, which means either one of the following two conditions:
 There exists a countable set such that for any , if and for all , then .
 There exists a countable set and there exists a continuous such that .
The map is the projection map from into the subproduct defined by . In Theorem 2, we only need to consider being defined on the subspace .
Theorem 2 follows from Theorem 1. It is only a matter of fitting Theorem 2 in the framework of Theorem 1. Note that the product is identical to the product where is a disjoint copy of the index set . For , let be defined by for all and for all .
With the identification of with , we have a setting that fits Theorem 1. The product is also a product of separable spaces. The set is a dense subspace of the product . In this new setting, we view a point in as . The map is still a continuous map. We can now apply Theorem 1.
Let be a countable set such that for all , if for all , then . Specifically, if for all and for all , then .
Choose a countable set such that and . Here, is the copy of in . We claim that is a countable set we need in condition 1 of Theorem 2. Let such that and for all . This implies that for all and for all . Then . Thus condition 1 of Theorem 2 holds. It is also straightforward to verify that Condition 1 and Condition 2 are equivalent.
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Remarks
The notion of factorizing a continuous map defined on a product space is an old topic. Theorem 1 discussed in this post is based on Theorem 4 found in [6]. Theorem 4 found in [6] is to factor continuous maps defined on a product of separable spaces. Theorem 1 in this post is modified to consider continuous maps defined on a dense subspace of a product of separable spaces. This modification will make it more useful. The references listed below represent a small sample of papers or books that have involves theorems of factoring functions defined on products. The work in [3] and [5] have more systematic treatment.
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Reference
 Brandenburg H., Husek M., On mappings from products into developable spaces, Topology Appl., 26, 229238, 1987.
 Engelking R., General Topology, Revised and Completed edition, Heldermann Verlag, Berlin, 1989.
 Engelking R., On functions defined on Cartesian products, Fund. Math., 59, 221231, 1966.
 Keesling J., Normality and infinite product spaces, Adv. in. Math., 9, 9092, 1972.
 Noble N., Ulmer M., Factoring functions on Cartesian products, Trans. Amer. Math. Soc., 163, 329339, 1972.
 Ross K. A., Stone A. H., Products of separable spaces, Amer. Math. Monthly, 71, 398403, 1964.
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