Let be a Tychonoff space (also called completely regular space). By we mean the space of all continuous real-valued functions defined on endowed with the pointwise convergence topology. In this post we discuss a scenario in which a function space can be embedded into another function space. We prove the following theorem. An example follows the proof.
Suppose that the space is a continuous image of the space . Then can be embedded into .
Proof of Theorem 1
Let be a continuous surjection, i.e., is a continuous function from onto . Define the map by for all . We show that is a homeomorphism from into .
First we show is a one-to-one map. Let with . There exists some such that . Choose some such that . Then since and .
Next we show that is continuous. Let . Let be open in with such that
where are arbitrary points of and each is an open interval of the real line . Note that for each , . Now consider the open set defined by:
Clearly . It follows that since for each , it is clear that .
Now we show that is continuous. Let where . Let be open with such that
where are arbitrary points of and each is an open interval of . Choose such that for each . We have for each . Define the open set by:
Clearly . Note that . To see this, let where . Now for each . Thus . It follows that is continuous. The proof of the theorem is now complete.
The proof of Theorem 1 is not difficult. It is a matter of notating carefully the open sets in both function spaces. However, the embedding makes it easy in some cases to understand certain function spaces and in some cases to relate certain function spaces.
Let be the first uncountable ordinal, and let be the successor ordinal to . Furthermore consider these ordinals as topological spaces endowed with the order topology. As an application of Theorem 1, we show that can be embedded as a subspace of . Define a continuous surjection as follows:
The map is continuous from onto . By Theorem 1, can be embedded as a subspace of . On the other hand, cannot be embedded in . The function space is a Frechet-Urysohn space, which is a property that is carried over to any subspace. The function is not Frechet-Urysohn. Thus cannot be embedded in . A further comparison of these two function spaces is found in this subsequent post.