The evaluation map is a useful tool for embedding a space into a product space and plays an important role in many theorems and problems in topology. See here for a previous discussion. In this post, we present the evaluation map with the perspective that the map can be used for embedding a space into a function space of continuous functions. This post will be useful background for subsequent posts on . In this post, we take a leisurely approach in setting up the scene. Once the map is defined properly, we show what additional conditions will make the evaluation map a homeomorphism. Then a function space perspective is presented as indicated above. After presenting an application, we conclude with some special cases for evaluation maps.

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**The general setting**

Let be a set (later we will add a topology). Let be a set of real-valued functions defined on . Another way to view is that it is a subspace of the product space . For each , consider the map defined by for all . One way to look at is that it is the projection map from the product space to one of the factors. Thus is continuous when has the product topology. In fact, the product topology is the smallest topology that can be defined on that would make the continuous. When we restrict the map to the subspace , the map is still continuous.

One more comment before defining the evaluation map. The set of functions is a subspace of the product space . Therefore the set inherits the subspace topology from the product space. It makes sense to consider the function space , the space of all continuous real-valued functions defined on endowed with the pointwise convergence topology. Thus we can write .

We now define the evaluation map. Define the map by letting for each , or more explicitly, by letting, for each , be the map such that for all .

The map is called the evaluation map defined by the family . When the set is understood, we can omit the subscript and denote the evaluation map by . We say is the canonical evaluation map.

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**What makes the evaluation map works**

The goal of the evaluation map is that it be a homeomorphism. For that to happen, we need to make a few more additional assumptions. In defining the evaluation map above, the functions in the family are not required to be continuous. In fact, the set is just a set in the above section. Now we require that is a topological space (it must be a completely regular space) and that all functions in are continuous. Thus we have . With this assumption, the evaluation map is then a continuous function. We have the following theorem.

*Theorem 1*

Let be a space. Let . Then the evaluation map is always continuous.

*Proof of Theorem 1*

Let . Let be open in with such that

where are arbitrary points of and each is an open interval of . For each , . Let , which is open in since each is a continuous function. We show that . For each and for each , . This means that for each , . The continuity of the evaluation map is established.

In order to make the evaluation map a homeomorphism, we consider two more definitions. A family is said to separate points of if for any with , there exists an such that . A family is said to separate points from closed subsets of if for each and for each closed subset of with , there exists an such that . We have the following theorem.

*Theorem 2*

Let be a space. Let . Then the following are true about the evaluation map .

- If separates the points of , then the evaluation map is a one-to-one.
- If separates the points from closed subsets of , then the evaluation map is a homeomorphism.

*Proof of Theorem 2*

To prove the bullet point 1, suppose that separates the points of . Let with . Then there is some such that . It follows that the functions and differ at the point . This completes the proof for the bullet 1 of Theorem 2.

To prove the bullet point 2, suppose that the family separates the points from closed subsets of . It suffices to show that the evaluation map is an open map. Let be a non-empty open set. We show that is open in image . Let where . Since separates the points from closed subsets of , there exists an such that . Let . Consider the following open set.

Clearly . We show that . Choose . Then for some . It is also the case that . Thus . This means that and that . This completes the proof for the bullet 2 of Theorem 2.

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**Embedding every space into a function space**

In defining the evaluation map, we start with a space . Then take a family of continuous maps . As long as the family of functions separates points from closed sets, we know for sure that the evaluation map is a homeomorphism from into a subspace of . We now look at some choices for . One is that . Then we have the following corollary.

*Corollary 3a*

Any space is homeomorphic to a subspace of the function space .

Because is a completely regular space, the family clearly separates points from closed sets. Thus Corollary 3a is valid. In fact, the complete regularity of only requires that we use , the set of all continuous functions from into where . We have the following corollary.

*Corollary 3b*

Any space is homeomorphic to a subspace of the function space .

We can also let , the set of all bounded real-valued continuous functions defined on . It is clear that separates points from closed sets. So we also have:

*Corollary 3c*

Any space is homeomorphic to a subspace of the function space .

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**One application**

We demonstrate one application of Corollary 3a. When is a separable metric space, has a countable network (see this previous post). It is natural to ask whether every space with a countable network can be embedded in a for some separable metric space ? The answer is yes. We have the following theorem. One direction of the theorem is Theorem III.1.13 in [1].

*Theorem 4*

Let be a space. Then the following conditions are equivalent.

- The space has a countable network.
- The space can be embedded in a for some separable metric space .

*Proof of Theorem 4*

The direction is clear. As shown here, has a countable network whenever has a countable base. Having a countable network carries over to subspaces. The direction is the one that uses evaluation map.

Suppose that is a countable network for . Then has a countable network, e.g., the set of all where , is any open interval with rational endpoints and is the set of all such that .

Any space with a countable network is the continuous image of a separable metric space. Thus there exists a separable metric space such that is the continuous image of . Let be a continuous surjection. Then can be embedded into . The embedding is defined by .

By Corollary 3a, is embedded into . Then is embedded into .

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**More on the evaluation map**

In this section, we consider some special cases. As shown in Theorem 2, what makes the evaluation map a one-to-one map is that the family separates points of (for short, the family is point separating). What makes the evaluation map a homeomorphism is that the family separates points from closed subsets of . In this section, we present one property that implies the property of separating points from closed sets. It is clear that if is dense in , then separates points of . In general, the fact that is point separating does not mean it separates point from closed sets. We show that whenever is compact, the fact that is dense in does imply that separates points from closed sets.

The family is said to be a generating set of functions if it determines the topology of , i.e., the following set is a base for the topology of .

Since is assumed to be a completely regular space, we observe that if is a base for , then the family separates points from closed subsets of . The following theorem captures the observations we make.

*Theorem 5*

Let be a space. Let . Then if is a generating set of functions, then separates points from closed subsets of , hence the evaluation map as defined above is a homeomorphism.

We now show that if is compact and if is dense in , then separates points from closed subsets of , making the evaluation map a homeomorphism.

First one definition. Let be a space. For any finite consisting of functions in , define the maximum of to be the function such that for each , is the maximum of the real values in . In other words, the maximum of is the pointwise maximum of the functions in . It is not too difficult to show that the pointwise maximum of finitely many continuous real-valued functions is also continuous. We have the following lemma and corollary.

*Lemma 6*

Let the space be compact. Suppose the family is dense in such that the pointwise maximum of any finite set of functions in is also in . Then separates points from closed subsets of .

*Proof of Lemma 6*

Let and let be a closed subset of such that . For each , consider the open set:

where is the open interval and is the open interval . For each , choose . The set of all is an open cover of the compact set . Choose such that cover where each . Let be the pointwise maximum of . By assumption, . It is clear that for all , . Thus .

Let . It is also clear that for all , implying . Thus . Thus completes the proof that separates points from closed subsets of .

*Corollary 7*

Let the space be compact. If is a dense subspace of , then the evaluation map as defined above is a one-to-one map.

In Corollary 7, even if is not closed under taking pointwise maximum of finitely many functions, then throw all pointwise maxima of all finite subsets of into and then apply Lemma 6. Throwing in all pointwise maxima will not increase the cardinality of . For example, suppose that is compact, is separable and is a countable dense subspace of . Even if does not contain all the pointwise maxima of finite subspaces, we can then throw in all pointwise maxima and the subspace is still countable. Then the compact space is homeomorphic to a subspace of . Since , is separable and metrizable. Thus the compact space is separable and metrizable. The following corollary captures this observation.

*Corollary 8*

If is a compact space and the function space is separable, then is metrizable.

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**Reference**

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

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