On real-valued homomorphisms in countably generated differential structures

Real valued homomorphisms on the algebra of smooth functions on a differential space are described. The concept of generators of this algebra is emphasized in this description.


Introduction
When all the real homomorphisms defined on an algebra of real functions defined on a space are evaluations then we say that the space is smoothly real-compact. There are many articles stating this property for various spaces. In [11], [4] it is shown that the spaces of real continuous functions on R and R n are smoothly real-compact. In [2] this property has been shown for the spaces of functions of class C k (k = 1, . . . , ∞) on separable Banach spaces. Much information about this topic can be found in [9]. The most important from the point of view of Sikorski spaces is the article [6] since it discusses smooth real-compactness of smooth spaces which are a wider category than the Sikorski spaces. Many conditions for those spaces to be smoothly real-compact are given there. In [1] and [7] many important results were obtained for a very wide class of algebras. In our article we emphasize the concept of generators of a differential space. We use techniques suitable for Sikorski spaces. Real valued homomorphisms are classified by their values on generators.

Basic concepts and definitions
Let M be a nonempty set and C a set of real functions on M . We introduce on M a topology τ C , the weakest topology in which the functions from C are continuous. We say that the set C is closed with respect to superposition if all functions of a form ω • (f 1 , . . . , f n ) where f 1 , . . . , f n ∈ C, ω ∈ C ∞ (R n ), n ∈ N, are in C. Adding to C all the functions of this form we obtain what we will call the superposition closure of C, denoted by scC following Waliszewski [15]. For any A ⊆ M by C A we denote the set of all functions f on A such that for any p ∈ A there exists an open neighborhood U ∈ τ C of p and a function g ∈ C such that f | U∩A = g| U∩A . If C = C M then we say that C is closed with respect to localization [13]. We call the set of real functions C on a nonempty set M a differential structure if it is: i) closed with respect to superposition, C = scC, ii) closed with respect to localization, C = C M . A differential structure is always an algebra with unity and contains all constant functions.

Definition 1.
A pair (M, C) is said to be a differential space if M is a nonempty set and C a differential structure on it.
We define a differential subspace of a differential space (M, C) to be any pair Thus C is the smallest differential structure that contains C 0 . Sometimes we write C = Gen C 0 . If C = Gen C 0 then for any f ∈ C and any point p ∈ M there exists an open neighborhood U ∈ τ C of p and functions f 1 , . . . , f n ∈ C 0 , ω ∈ C ∞ (R n ), n ∈ N such that f | U = ω • (f 1 , . . . , f n )| U . We say that the differential space (M, C) is finitely generated if it is generated by a finite set of a real functions. A differential space is countably generated if it is generated by a countable set of real functions but it is not finitely generated.
We denote by (R I , ε I ) the differential space with the structure ε I generated by the set of projections C 0 = {π i : i ∈ I}, where π i : R I → R is defined by π i (x) = x i for x = (x i ) i∈I . This is a generalization of the Cartesian space (R n , ε n ) where ε n = C ∞ (R n ).
The spectrum of an algebra C is the set where χ is a homomorphism that preserves unity. Evaluation of the algebra C at a point p ∈ M is the homomorphism χ ∈ Spec C given by We will denote it by ev p . We define the mapping ev : M → Spec C by the formula: Definition 3. ( [6]) We say that a differential space (M, C) is smoothly real-compact if any χ ∈ Spec C is evaluation at some point p ∈ M .
From this definition it follows that the space (M, C) is smoothly real-compact when the mapping ev is a surjection. For any f ∈ C we define the functionf : The set of all functions of the formf will be denoted byĈ. Define τ : C →Ĉ by The mapping τ is an isomorphism between the algebra C and the algebraĈ.

Main results
Lemma 1. The differential space (R n , ε n ) is smoothly real-compact.
Proof. Let χ ∈ Spec ε n . We define p ∈ R n by p i := χ(π i ) for i = 1, . . . , n. We will show that χ = ev p . It is known that any f ∈ ε n can be represented as where the functions g i satisfy g i (p) = ∂ i f (p). Then . Therefore χ(f ) = f (p) for all f ∈ ε n . Now we prove: Lemma 2. Every differential subspace of the differential space (R n , ε n ) is smoothly real-compact.
Proof. Let (M, C) be a differential subspace of (R n , ε n ). The inclusion mapping ι M : M → R n is smooth and therefore ι * M : ε n → M is a homomorphism. From the definition we know that ι * M (f ) = f | M for all f ∈ ε n . For any χ ∈ Spec C we have χ • ι * M ∈ Spec ε n . From Lemma 1 we know that there exists p ∈ R n such that Since ω| M > 0 we have 1 ω|M ∈ C. We also know that This is a contradiction.
We will show that χ = ev p . Let f ∈ C.
There exists an open neighborhood U ∈ τ εn of p and a function κ ∈ ε n such that f | U∩M = κ| U∩M . From [14] we know that there exists a bump function φ ∈ ε n with φ(p) = 1, φ| M∩U > 0 and If the differential structure C of the differential space (M, C) is generated by a set of functions C 0 then we can define a mapping φ : We will call this mapping the generator embedding. We can prove the following: Proof. If C 0 separates the points of M then φ is a diffeomorphism onto its image so the result is obvious. So assume that C 0 does not separate points. Thenφ : From the last lemma we know that it is sufficient to work on subspaces of Cartesian spaces. Corollary 1. Let (M, C) be a differential space with C = Gen C 0 for some finite C 0 . Then (M, C) is smoothly real-compact.
From Corollary 1 we obtain: Lemma 4. Let (M, C) be a differential space. Any χ ∈ Spec C satisfies the following condition: for all ω ∈ ε n and f 1 , . . . , f n ∈ C, n ∈ N.
Proof. To prove that (Spec C,Ĉ) is a differential space, we have to show that the set C is closed with respect to superposition with smooth functions from ε n and is closed with respect to localization.
Let a function f : Spec C → R satisfy the localization condition in the space (Spec C,Ĉ). For any open subsetÛ ∈ Spec C there isĝ ∈Ĉ such that f |Û =ĝ|Û . We can uniquely define a function h : M → R by the condition h(p) = f (ev p ) for all p ∈ M . For any open setÛ ∈ Spec C the set U = {p ∈ M : ev p ∈Û } is open. From the definitions of h and U we know that h| U = g| U . Because g ∈ C it follows that h ∈ C. We also know thatĥ| evM = f | evM . From Corollary 3 we derive that f =ĥ. This means that f ∈Ĉ soĈ is closed with respect to localization. Now one can prove the following lemmas: Lemma 7. If (M, C) is a differential space with the structure C generated by C 0 then the differential structureĈ of the differential space (Spec C,Ĉ) is generated byĈ 0 .
Proof. Assume that C 0 = {f i : i ∈ I}. We know that for any f ∈ C there exists an open covering of M such that on each set U of this covering the function f can be expressed in the form ω • (f 1 , . . . , f n ) where f 1 , . . . , f n ∈ C and ω ∈ ε n . For each open set U of the covering we defineÛ = {ev p ∈ Spec C : p ∈ U }. On the setÛ we consider the functionf = τ (ω • (f 1 , . . . , f n )). The sets of the formÛ might not be a covering of Spec C but thair union is dense in Spec C. Therefore we can prolong uniquely this representation off on the whole Spec C. We have shown that C = GenĈ 0 . Lemma 8. For any differential space (M, C) the differential space (Spec C,Ĉ) is smoothly real-compact.
Proof. We definef (p) =f (χ) where χ ∈ Spec C is such that χ(π i ) = p i for all i ∈ I. Since a homomorphism is uniquely defined by its value on the generators (Lemma 9) this definition is correct. We see that if p ∈ M then χ = ev p andf (p) =f (ev p ) = f (p) so this is indeed a prolongation. This prolongation is continuous since the functioñ f is a realization of the functionf on the setM which is the image of Spec C under the generator embedding using the generators τ (π i | M ), i ∈ I. Uniqueness follows from the fact that M is dense inM in the topology of R I . From Lemma 10 we obtain: Corollary 4. When (M, C) is a differential subspace of (R I , ε I ) generated by C 0 = {π i | M : i ∈ I} then the mapping χ : C 0 → R defined on generators byχ(π i | M ) = p i for some p ∈M − M can be prolonged to a homomorphism on the whole C iff all the functions from C are prolongable to p.
Let M = R N − {0} and C M = (ε N ) M . Then (M, C M ) is a differential subspace of (R N , ε N ). We will show that this space is smoothly real-compact. Lemma 11. There exists a function ξ ∈ C M which is not prolongable to any continuous function on R N .
Proof. We know that there exists a function φ ∈ C ∞ (R) satisfying the following properties: We will show that this function belongs to the structure C M . For any k ∈ N we can define the closed subset For any (x n ) ∈ M the sequence ρ k ((x n )) is non-decreasing with respect to k and there exists k 0 ∈ N for which 1 k 2 < ρ k0 ((x n )). This means that (x n ) / ∈ A k . Therefore k∈N A k = ∅. We also know that A k+1 ⊆ A k . Let us define the family of open subsets U k = M − A k . Of course k∈N U k = M . If (x n ) ∈ U k then φ(k 2 ρ k ((x n )) = 0. Then for all m > k, x n ∈ U m so φ(m 2 ρ m ((x n ))) = 0. This means that only a finite number of elements are non-zero in the sum (9) and therefore From the localization closedness of the differential structure we derive that ξ ∈ C M . Now we will define a sequence in M convergent to 0 on which the function ξ diverges. Let z k = (x n,k ) where We can see that lim k→∞ z k = 0 ∈ R N and for j < k.
For j ≤ k we obtain φ(j 2 ρ j ((x k ))) = 1 and therefore This means that lim k→∞ ξ((x k )) = +∞. The function ξ is not prolongable to any continuous function in R N .

Now we prove
Lemma 12. The differential space (M, C M ) is smoothly real-compact.
Proof. From Lemma 9 we know that the set Spec C M may contain only one homomorphism χ 0 which is not an evaluation. This homomorphism would be defined on the generators by the formula χ 0 (π i | M ) = 0 for all i ∈ I. So there would be only one point 0 ∈M − M . But it cannot be so since from Corollary 4 we know that all the functions from C M are prolongable to the point 0. From the last lemma we know that there exists a function ξ ∈ C M which is not prolongable.
One can easily see Proof. Let ι M : (M, C) → (R N , ε N ) be the inclusion mapping. For any χ ∈ Spec C, We need to show that p ∈ M . Assume that p / ∈ M . We can treat the space (M, C) as a differential subspace of (R N ,ε). Let ν M : (M, C) → (R N ,ε) be the inclusion. Then χ • ν * M ∈ Specε. Because the space (R N ,ε) is smoothly real-compact there exists a point q ∈ R N such that χ • ν * M = ev q |ε. We know that on common generators π i the equalities χ(π i | M ) = ev p (π i ) = p i and χ(π i | M ) = ev q (π i ) = q i holds for all i ∈ N. This specifies all the coordinates so p = q. Therefore we can write We have a contradiction with the fact that (χ • ν * M )(θ p ) = χ(θ p | M ) = χ(0) = 0. We see that p ∈ M and χ • ι * M = ev p | ε N . So χ(π i | M ) = ev p (π i |M ) for all i ∈ N. The set {π i : i ∈ N} is the set of generators of the differential space (M, C). We derive that χ = ev p . Corollary 6. Any countably generated differential space is smoothly real-compact.
Proof. Every countably generated differential space can be treated as a subspace of (R N , ε N ). From Theorem 1 we know that all subspaces of this space are smoothly real-compact.

Conclusion
We have shown how a real-valued homomorphism act on the algebra of smooth functions in a differential space. It is sufficient to examine its value on the generators of the differential structure. A non-prolongable function on a differential space of sequences without zero sequence is constructed. From the existence of this function we deduced that countably generated structural algebra C of a differential space (M, C) gives all information about the set M because the set Spec C contains only real valued homomorphisms of the form ev p for p ∈ M . Therefore the geometry of such spaces can be built on their algebras. The choice of generators of the differential structure is not unique so it is important to choose the smallest one. It is also shown that the pair (Spec C,Ĉ) is a differential space for any differential space (M, C). For countably generated differential spaces, (M, C) and (Spec C,Ĉ) are diffeomorphic. Theorem 1 has been obtained as a result of observations about generators. From [1] and [7] it also follows as a conclusion from a much wider theory. In [12] there is an important theorem which gives a sufficient condition for an algebra of functions on an arbitrary topological space to be smoothly real-compact. We could easily show that countably generated differential spaces satisfy this condition. But we have presented our proof using techniques of differential space theory.