Smooth embeddings of the Long Line and other nonparacompact manifolds into locally convex spaces
Abstract
We show that every real finite dimensional Hausdorff (not necessarily paracompact, not necessarily second countable) manifold can be embedded into a weakly complete vector space, i.e. a locally convex topological vector space of the form for an uncountable index set and determine the minimal cardinality of for which such an embedding is possible.
Keywords: nonparacompact manifolds, long line, locally convex space, weakly complete space
MSC2010: 57R40 (primary); 46T05, 46A99 (Secondary)
1 Introduction and statement of the results
We review the classical Theorem of Whitney (see e.g. [Whi36, Theorem 1], [AM77, Theorem 6.3], or [Ada93, Theorem 2.2]):
Theorem 0. (Whitney)
Let be a dimensional second countable Hausdorff manifold (). Then there exists a embedding into .
The conditions (Second Countability and the Hausdorffproperty) are obviously also necessary, since every Euclidean space is second countable and Hausdorff and so are all of its subsets. The dimension in the Theorem is sharp in the sense that whenever is a power of two, i.e. , there is a dimensional second countable Hausdorff manifold which can not be embedded into .
Unfortunately, not every Hausdorff manifold is second countable. Perhaps the easiest connected manifold where the second axiom of countability does not hold, is the Long Line and its relative the Open Long Ray (see e.g. [Kne58]). For the reader’s convenience, we will recall the definition:
Definition 1.1. (The Alexandroff Long Line)

Let be the first uncountable ordinal. The product , endowed with the lexicographical (total) order, becomes a topological space with the order topology, called the Closed Long Ray. This space is a connected (Hausdorff) onedimensional topological manifold with boundary .

To obtain a manifold without boundary, one removes this boundary point. The resulting open set is called the Open Long Ray.

A different way to obtain a manifold without boundary is to consider two copies of the Closed Long Ray and glue them together at their boundary points. The resulting onedimensional topological manifold is called the Long Line^{2}^{2}2Some authors use the term Long Line for what we call here Long Ray..
The spaces and are locally metrizable but since countable subsets are always bounded, none of the three spaces is separable. So, in particular, they are not second countable. Since we are only considering manifolds without boundary in this paper, for us only the Open Long Ray and the Long Line are interesting.
It is known that there exist structures on and on for each . They are however not unique up to diffeomorphism. For example, there are pairwise nondiffeomorphic structures on (cf. [Nyi92]).
The Long Line and the Open Long Ray are by far not the only interesting examples of nonsecond countable manifolds. A famous two dimensional example (which has been known even before the Long Line) is the so called Prüfer manifold (see [Rad25] for a definition).
Since these manifolds fail to be second countable they cannot be embedded into a finite dimensional vector space. However, one can ask the question if it is possible to embed them into an infinite dimensional space. Of course, before answering this question, one has to say concretely what this should mean as there are different, nonequivalent notions of differential calculus in infinite dimensional spaces: We use the setting of MichalBastiani, based on Keller’s calculus (see [Glö02], [Kel74], [Mil84] and [Nee06]). This setting allows us to work with maps between arbitrary locally convex spaces, as long as they are Hausdorff. A manifold modeled on a locally convex space can be defined via charts the usual way, there is a natural concept of a submanifold generalizing the concept in finite dimensional Euclidean space. Important examples of locally convex spaces are Hilbert spaces, Banach spaces, Fréchet spaces and infinite products of such spaces such as for an arbitrary index set . Since this differentiable calculus explicitly requires the Hausdorff property, we will not consider embeddings of nonHausdorff manifolds, although there are interesting examples of those (occurring naturally as quotients of Hausdorff manifolds, e.g. leaf spaces of foliations etc.) Our first result is the following:
Theorem A 0. ()
Let . Let be a finite dimensional Hausdorff manifold (not necessary second countable). Then there exists a set such that can be embedded into the locally convex topological vector space .
A locally convex vector space of the type is called weakly complete vector space (see [BDS15, Appendix C] or [HM07, Appendix 2]). These weakly complete spaces form a good generalization of finite dimensional vector spaces. The cardinality of the set , sometimes called the weakly complete dimension, is a topological invariant of (see Lemma 3.1). This gives rise to the following question: Given a finite dimensional manifold , what is the minimal weakly complete dimension which is necessary to embed ? Unfortunately, we will not be able to answer this question completely, but we will give upper and lower bounds and if the Continuum Hypothesis is true, then we have a complete answer:
Theorem B 0. ()
Let be finite dimensional Hausdorff manifold (not necessary second countable). The embedding dimension of is defined as
Furthermore, let denote the number of connected components of . Then the following holds:

The cardinal is finite if and only if is second countable.

If is not second countable and (the continuum), then

If , then
In particular, is never equal to .
Of course, if the continuum hypothesis holds, then this theorem gives an exact answer.
Now for something completely different: By a theorem of Kneser (see [Kne58]), there is a realanalytic () structure on the Long Line. (in fact, there are infinitely many nonequivalent of them (see e.g. [KK60, Satz 1])). For second countable manifolds, there is an analogue of Whitney’s Theorem for real analytic manifolds, stating that every second countable Hausdorff manifold can be embedded analytically into a Euclidean space (see e.g.[For11, 8.2.3]). This rises the question whether we can embed a real analytic Long Line into a space of the form . We answer this question in the negative:
Theorem C 0. ()
Let be the Long Line or the Open Long Ray with a structure. Then it is not possible to embed as a submanifold into any locally convex topological vector space. In fact, every map from into any locally convex topological vector space is constant.
Lastly, we will address the question whether the embeddings we constructed in Theorem A have closed image. It is a wellknown (and easy to show) fact that each submanifold of is diffeomorphic to a closed submanifold of . Hence, every submanifold of a finite dimensional space can be regarded as a closed submanifold of a (possibly bigger) finite dimensional space^{3}^{3}3It is also possible to construct the Whitney embedding already in such a way that the image is closed in .. The question whether this also holds in our setting, i.e. whether the Long Line is diffeomorphic to a closed submanifold of is answered to the negative:
Theorem D 0. ()
Let be the Long Line or the Open Long Ray and let be any complete locally convex vector space. Then is not homeomorphic to a closed subset of . In particular, cannot be a closed submanifold of .
If one allows noncomplete locally convex spaces, then there is a way to embed topologically as a closed subset into a noncomplete locally convex space (see Remark 5.5). However, this embedding is merely continuous but fails to be . It is not known to the author if there is a way to construct a embedding with a closed image in a locally convex space.
2 Construction of the embedding
2.1. ()
Whenever we speak of mappings and manifolds, we refer to the locally convex differential calculus by MichalBastiani. Details can be found in [Glö02], [Kel74], [Mil84] and [Nee06]. For the special case that a function is defined on , this is equivalent to the notion that is times partially differentiable and that all are continuous (see for example [Wal12, Appendix A.3]) All manifolds are assumed to be Hausdorff but we do not assume that they are connected (and of course we will not assume that they are paracompact or even second countable).
2.2. (submanifolds)
Let . Let be a manifold modeled on a locally convex space and be a closed vector subspace. A subset is called a submanifold of modeled on if for each point there is a diffeomorphism with and open such that .
The submanifold then carries a natural structure of a manifold modeled on the vector space . It should be noted that although is assumed to be closed in , we do not assume that is a closed subset of .
2.3. (embeddings)
Let and be locally convex manifolds and let be a map. We call a embedding if is a submanifold of and is a diffeomorphism. Our goal is to show that for each finite dimensional there is a set such that there is a embedding .
2.4. (Immersions)
One reason why locally convex differential calculus is more involved than in finite dimensions is that there are at least two different notions of immersions: Let be a map between locally convex manifolds. For the sake of this article, let us call a weak immersion if the tangent map at each point is injective. We call a strong immersion if every has an open neighborhood such that is a embedding.
Lemma 2.5. (Immersion Lemma)
For a map on a finite dimensional manifold and a locally convex manifold , the following are equivalent:

is a weak immersion.

is a strong immersion.
Proof.
Since both properties are local, we may assume that is a map, where is an open subset of , while is a (Hausdorff) locally convex topological vector space.
Since the implication (b) (a) holds trivially, even without the domain being finite dimensional, we will show (a)(b). To this end, let be fixed. We assume that is injective. This means that is a dimensional vector subspace of . Since finite dimensional vector subspaces in locally convex spaces are always complemented (see e.g. [Jar81, Corollary 2 in Chapter 7.2]), we may assume that with a closed vector subspace . We obtain the projections and . It is easy to check that the tangent map of is an invertible linear map between the dimensional real vector spaces and . Hence, by the usual Inverse Function Theorem for maps, there exists a small neighborhood of such that maps diffeomorphic onto . The inverse map will be denoted by .
Now, the function is a map and hence, its graph
is a submanifold of . It is now easy to check that the image of is the set and that the map is a diffeomorphism onto its image. ∎
Remark 2.6. ()
While this Lemma holds for finite dimensional , it has to be said that statements like these fail to hold if the domain is infinite dimensional, in particular beyond Banach space theory due to the lack of an Inverse Function Theorem.
This main essence of this last lemma (with slightly different definitions and vocabulary) can also be found in [Glö15] which provides a good overview of immersions and submersions in infinite dimensional locally convex spaces.
Proposition 2.7. ()
Let be a map between locally convex manifolds and . Then is a embedding if and only if is a topological embedding and a strong immersion.
Proof.
Having the right (strong) definition of immersion, this proposition is easy to show: Let . Then there exists an open neighborhood of such that is a embedding. Since is a topological embedding, the image of under is open in ,i.e. there exists an open neighborhood of in such that . ∎
Now, we are ready to show that every finite dimensional Hausdorff manifold admits a embedding into a locally convex space of the type for an index set :
Proof of Theorem A.
Let be a finite dimensional manifold and let
denote the set of all compactly supported functions on . We define the following map
This map is since every component is , in particular, it is continuous.
Next, we show that it is a topological embedding. Let be a point and let be a net in with the property that converges to . We will show that converges to . To this end, let be an open neighborhood of . It is possible to construct a function such that and . Since converges to in the product space and since projection onto the th component is continuous, we obtain that converges in to . Hence, there is an such that for all . This implies that for all and hence, is a topological embedding.
It remains to show that satisfies part (a) of Lemma 2.5. Then, together with Proposition 2.7, the assertion follows.
To this end, let be fixed. It remains to show that the linear map is injective. Since this is a local property, we may assume that is an open neighborhood in and that .
We obtain the following formula for the linear map:
Let and fix a linear map . We define the following function via
where is a suitable function with compact support in and the property that for all in a small neighborhood of . Since , this implies that for all . In particular, we have that . But since is equal to in a neighborhood of , this implies that . Since was arbitrary, this implies that . This finishes the proof. ∎
3 The embedding dimension
In this section we will give a proof of Theorem B stated in the introduction.
Lemma 3.1. (Weight = Weakly Complete Dimension)
Let be a weakly complete vector space with infinite. Then the cardinality of is a topological invariant of the space, i.e. it can be computed using only the topology of and not the vector space structure:

The cardinal is the minimal cardinality of a basis of the topology of , i.e. is the weight of the topological space .

The cardinal is the maximal cardinality of a discrete subset of the space .
Proof.
If a topological space has a topological basis of cardinality at most . Then each subset has the same property. In particular, each discrete subset has a topological basis of cardinality at most which implies that the discrete subset itself has at most many elements. This shows that the maximal cardinality of a discrete subset is less than or equal to the minimal cardinality of a basis.
The product topology on has a basis of the topology of many sets (using that is infinite). This shows that the minimal cardinality of a basis is bounded above by .
Lastly, the set is discrete, showing that is less than or equal to the maximal cardinality of a discrete subset. Putting these arguments together, the claim follows. ∎ of unit vectors in
Remark 3.2. ()
In the case that is finite, the weakly complete dimension of is no longer equal to the weight of the spaces . However, is still uniquely determined by the topology of by the invariance of dimension from algebraic topology (see e.g. [Hat02, Theorem 2.26]). However, we will not need this fact here.
We will start with a lemma which can be found in [Cla83, Theorem (i)]
Lemma 3.3. ()
Let be a finite dimensional topological connected Hausdorff manifold. Then admits an atlas of continuum cardinality.
Using this lemma, we can easily proof the following:
Lemma 3.4. ()
Let be a finite dimensional topological connected Hausdorff manifold. Then admits an open cover which is stable under finite unions and such that each is separable and such that .
Proof.
By Lemma 3.3 we know that has an atlas with . Every chart domain of is homeomorphic to a subset of and hence separable. Unfortunately, the union of two chart domains is in general not a chart domain. Hence, we consider all finite unions of chart domains. The finite union of open separable sets is open and separable. If the cardinality of the atlas is infinite, it will not increase by allowing finite unions of elements. Hence, it will still be bounded above by the continuum. ∎
We will now give the proof of Theorem B:
Proof of Theorem B.
Part (a) is just the classical Theorem of Whitney for maps stated in the introduction.
For the proof of part (b), assume that is not second countable. Since is a separable Fréchet space it is second countable. So, cannot be homeomorphic to a subset of . Hence, .
Assume now that , i.e. is connected. Let be the open cover from Lemma 3.4. For each , let be the space of all such that . Every support of a function is compact and hence can be covered by finitely many . Since the system is directed, we may conclude that for each there is a such that , i.e.
Now, every is separable, i.e. there is a dense countable set . Each function is in particular continuous and hence uniquely determined by its values on the dense subset , yielding an injective map
In the proof of Theorem A, we saw that can be embedded in with . This allows us to estimate the embedding dimension of as follows:
This finished the proof for the case that .
Now, for case : Let be the family of connected components of . By the preceding calculation, we know that each admits a embedding
For each , we define the function
As is a function from the index set to , we have that belongs to the weakly complete vector space . It is easy to see that . is a discrete subset of
Now, we are able to define the embedding of :
Is it straightforward to check that this is a embedding. Using this embedding, we obtain an upper bound for the embedding dimension:
where the last equality used the wellknown fact in cardinal arithmetic that the product of two cardinals is equal to the maximum if both are nonzero and at least one of them is infinite.
It remains to show that . To this end, we chose from each connected component one element . Then it is easy to see that the set . This implies that has a discrete subset of cardinality . By Lemma 3.1, the cardinality of a discrete subset of a weakly complete space is always bounded above by the weakly complete dimension of the surrounding space. Hence . This finishes the proof of Theorem B. ∎ is discrete in
4 Analytic embeddings
In this section we will give a proof of Theorem C stated in the introduction. To this end, let be either the Long Line or the Open Long Ray, together with one of the structures on it. Let be any locally convex space and consider a map . We will show that is constant. To this end, let be any continuous linear functional on . Continuous linear maps are always analytic, so are compositions of real analytic maps (see [Glö02, Proposition 2.8]). This implies that is a function on . However, a wellknown fact about the Long Line (and the Open Long Ray) is that every continuous function becomes eventually constant (see e.g..[Nyi92, Theorem 7.7]). So, from a point onwards, will be constant and by the Identity Theorem for analytic functions (and the fact that is connected), this implies that is globally constant on .
Since the functional was arbitrary and by HahnBanach, the continuous linear functionals separate the points of , it follows that is constant. So in particular, cannot be an embedding.
5 Closed embeddings
In this section we will give a proof of Theorem D stated in the introduction. Only for internal use in this article, we will use the following terminology:
Definition 5.1. ()
A topological space is called a space if every closed sequentially compact subset is compact.
Clearly, a closed subset of a space is again .
Lemma 5.2. ()
The Long Line and the Open Long Ray are not spaces.
Proof.
Take an element in the Open Long Ray and consider the set of all elements . Then is closed and sequentially compact but not compact. Hence, the Long Ray is not .
The Long Line is closed in itself and sequentially compact but not compact. Thus, it is not . ∎
We will now show that a complete locally convex space always has property . Then Theorem D follows immediately.
Lemma 5.3. ()
A locally convex topological vector space is if at least one of the following conditions is satisfied:

is metrizable

is complete

is Montel
Proof.
If is metrizable, then every subset is metrizable. Hence, sequentially compactness is equivalent to compactness.
Let be Montel and let be a closed sequentially compact subset. Then is compact for every continuous seminorm . Hence, is bounded. But closed and bounded subsets of Montel spaces are compact.
Let be a complete locally convex space and let be closed and sequentially compact. Every locally convex space is isomorphic to a vector subspace of a product of Banach spaces. Thus, we may assume that
where each is a Banach space. The projection is continuous, hence is sequentially compact in . Since is metrizable, each is compact. The set is now contained in the product which is compact by Tychonoff. Now, is closed in and (since is complete) is closed in the product, hence is closed in and contained in the compact set . Thus, is compact. ∎
Remark 5.4. ()
Recall that a finite dimensional manifold is called bounded, if every countable subset is relatively compact. Since every bounded finite dimensional manifold is sequentially compact, the exact same argument as above shows that every noncompact bounded manifold fails to be and hence cannot be embedded as a closed subset of a complete locally convex vector space.
Remark 5.5. ()
If one allows locally convex spaces which are not , then we can embed every finite dimensional manifold topologically as a closed subset.
Fix a manifold and consider the set where denotes the constant function on . Then the construction in the proof of Theorem A yields a embedding:
Now, let be the real vector subspace of generated by the image of . One can verify that is closed in and since carries the subspace topology of , the map is still a topological embedding. This shows that can always be embedded as a closed subset in a locally convex vector space.
Unfortunately, this map will no longer be as a map with values in the nonclosed subspace (although it is as a map with values in the surrounding space ). Hence, this construction does not give us a embedding of into .
It is not known to the author if there is a different construction such that embeds as a closed submanifold in a locally convex space.
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