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.
Theorem 1
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.
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Example
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.
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