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On differential completions and compactifications of a differential space

Differential completions and compactifications of differential spaces are introduced and investigated. The existence of the maximal differential completion and the maximal differential compactification is proved. A sufficient condition for the existence of a complete uniform differential structure on a given differential space is given.


Introduction
This article is the third of the series of papers concerning integration of differential forms and densities on differential spaces (the first two are [4] and [5]). We describe differential completions and differential compactifications of differential spaces which are used in our theory of integration. Section 2 of the paper contains basic definitions and the description of preliminary facts concerning theory of differential spaces. In Section 3 we give basic definitions and describe the standard facts concerning theory of uniform spaces. We introduce the notion of a differential completion of a differential space. We construct differential completions of a differential space using families of generators of its differential structure (Proposition 3.7, Definition 3.12). Section 4 is devoted to the investigation of properties of differential completions. We define some natural order in the set of all differential completions of a given differential space. We prove that for any differential space (M, C) there exists the maximal differential completion with respect to this order (Theorem 4.1). If for the uniform structure defined by some family of generators the space M is complete then the appropriate differential completion of (M, C) is maximal and coincides with (M, C) (Theorem 4.2). At the end we prove that if a differential structure C posses a countable family of generators then it coincides with its maximal differential completion (Theorem 4.3). As a corollary we obtain general topological result about the existence on a given topological space a uniform structure defining the initial topology (Corollary 4.1). In Section 5 we introduce and investigate the notion of a differential compactification of a differential space. Similarly as in Section 4 we prove the existence of the maximal differential compactification of a given differential space with respect to the suitable order.
Without any other explanation we use the following symbols: N-the set of natural numbers; R-the set of reals.

Differential spaces
Let M be a nonempty set and let C be a family of real valued functions on M . Denote by τ C the weakest topology on M with respect to which all functions of C are continuous.
The set of all C-smooth functions on M is denoted by scC.
Since C ⊂ scC and any superposition ω • (α 1 , . . . , α n ) is continuous with respect to τ C we obtain τ scC = τ C (see [4], [5]). DEFINITION 2.3 A set C of real functions on M is said to be a (Sikorski's) differential structure if: (i) C is closed with respect to localization i.e. C = C M ; (ii) C is closed with respect to superposition with smooth functions i.e. C = scC.
In this case a pair (M, C) is said to be a (Sikorski's) differential space (see [8]). Any element of C is called a smooth function on M (with respect to C).
PROPOSITION 2.1. The intersection of any family of differential structures defined on a set M = ∅ is a differential structure on M .
Let F be a set of real functions on M . Then, by Proposition 2.1, the intersection C of all differential structures on M containing F is a differential structure on M . It is the smallest differential structure on M containing F . One can easy prove that C = (scF ) M (see [9]). This structure is called the differential structure generated by F and is denoted by gen(F ). Functions of F are called generators of the differential structure C. We have also τ (scF )M = τ scF = τ F (see remarks after Definitions 2.1 and 2.2).
Let (M, C) and (N, D) be differential spaces. A map F : M → N is said to be smooth if for any β ∈ D the superposition β • F ∈ C. We will denote the fact that F is smooth writing F : (M, C) → (N, D).
If A is a nonempty subset of M and C is a differential structure on M then C A denotes the differential structure on A generated by the family of restrictions {α |A : α ∈ C}. The differential space (A, C A ) is called a differential subspace of (M, C). One can easy prove the following If the map F : (M, C) → (F (M ), F (M ) D ) is a diffeomorphism then we say that F : M → N is a diffeomorphism onto its range (in (N, D)). In particular the natural embedding A ∋ m → i(m) := m ∈ M is a diffeomorphism of (A, C A ) onto its range in (M, C).
If {(M i , C i )} i∈I is an arbitrary family of differential spaces then we consider the Cartesian product i∈I M i as a differential space with the differential structureˆ i∈I C i generated by the family of functions F : The topology τˆ i∈I Ci coincides with the standard product topology on i∈I M i . We will denote the differential structureˆ i∈I C ∞ (R) on R I by C ∞ (R I ). In the case when I is an n-element finite set the differential structure C ∞ (R I ) coincides with the ordinary differential structure C ∞ (R n ) of all real-valued functions on R n which posses partial derivatives of any order (see [8]). In any case a function α : R I → R is an element of C ∞ (R I ) iff for any a = (a i ) ∈ R I there are n ∈ N, elements i 1 , i 2 , . . . , i n ∈ I, a set U open in R n and a function ω Let F be a family of generators of a differential structure C on a set M . The generator embedding of the differential space (M, C) into the Cartesian space defined by F is a mapping φ F : (M, C) → (R F , C ∞ (R F )) given by the formula If F separates points of M the generator embedding is a diffeomorphism onto its image. On that image we consider a differential structure of a subspace of (R F , C ∞ (R F )) (see [5], Proposition 2.3).
Let M be a group (a ring, a field, a vector space over the field K). A differential structure C on M is said to be a group (ring, field, vector space) differential structure if the suitable group (ring, field, vector space) operations are smooth with respect to C, C ⊗ C and C K , where C K is a field differential structure on K. In this case the pair (M, C) is called a differential group (ring field, vector space). If K = R or K = C we take C K = C ∞ (K) as a standard field differential structure. PROPOSITION 2.3. Let V be a vector space over R and let F be a family of constant functions and linear functionals defined on V . Then the differential structure C generated by F on V is a vector space differential structure.
For the proof see [4], Proposition 2.3. The set T M can be endowed with a differential structure in the following standard way. We define the projection π : T M → M such that for any m ∈ M and any v ∈ T m M π(v) = m.
For any α ∈ C we define the differential (or the exterior derivative) of α as a map dα : T M → R given by the following formula Then we define T C as the differential structure on T M generated by the family of functions T C 0 := {α • π : α ∈ C} ∪ {dα : α ∈ C}. From now on we will consider T M as a differential space with the differential structure T C.
For any m ∈ M we will denote by dα m the restriction dα |TmM . It is clear that dα m is a linear functional on T m M .
We have also that π : (T M, T C) → (M, C). Then π is continuous and for any . It can be proved that T U is (isomorphic to) a tangent space to the differential space (U, C U ).
is  (1) is called the map tangent to F .
Proof. The second part of the thesis follows immediately from Proposition 2.4 because for any . For the proof of the first part of the thesis it enough to show that for any κ ∈ T D 0 the superposition κ • T F ∈ T C. Let us consider the case κ = β • π N . We have Let us consider the differential space (R I , C ∞ (R I )). The differential structure C ∞ (R I ) is generated by the family of projections F := {pr i } i∈I , where For any is well defined (in some neighbourhood of x the function α depends on finite number of variables x i ) and is a vector tangent to R I at x. On the other hand, if u ∈ T x R I and for any i ∈ I we denote v i := u(pr i ) then for any Then we identify the set T x R I with {x} × R I . Consequently we identify the set T R I with R I × R I . In this case the differential structure T C ∞ (R I ) is generated by the family of functions Hence for any j ∈ I It means that T C ∞ (R I ) = C ∞ (R I × R I ) and consequently for any x ∈ R I the differential structure T C ∞ (R I ) TxR I is generated by the family of projections {pr ′ i : Let φ F : (M, C) → (R F , C ∞ (R F )) be the generator embedding of the differential Hausdorff space (M, C) defined by some family of generators F . Then we can identify differential spaces (M, C) and THEOREM 2.2. Let I be an arbitrary nonempty set and let X be a nonempty subset of the Cartesian space R I . Then for any For the proof see [4], Theorem 3.2.
A map X : M → T M such that for any m ∈ M the value X(m) ∈ T m M is called a vector field on M . A vector field X on M is smooth if X : (M, C) → (T M, T C).

Uniform structures and completions of a differential space defined by families of generators
For the general theory of uniform structures and completions see [6], Chapter 8 or [1]. It is also described in [7] and [5]. Here we start with the definition of the uniform structure given on a differential space by a family F of generators of its differential structure.
Let F be a family of real-valued functions on a set M and let (M, C) be a differential space such that C = (scF ) M and (M, τ C ) is a Hausdorff space (the last is true iff the family C separates points in X iff the family F separates points in X). On the set M the family F defines the uniform structure U F such that the base B of U F is given as follows: where (see [5], Proposition 3.1).
The uniform structure U on a set M is said to be a differential uniform structure on the differential space (M, C) if there exist a family F of generators of C such that U = U F , where U F is defined by the base (2). The uniform space (M, U F ) is said to be the uniform space given by the family of generators F .
If we have two different families F 1 and F 2 of generators of a differential space (M, C), then the uniform structures U F1 and U F1 can be different too.
Then does not exists ε > 0 such that If F is a family of generators of a differential structure C on a set M then we define a uniform structure U T F on the space T M tangent to the differential space (M, C) using the family of real-valued functions where π : T M → M is the natural projection and df : T M → R, df (v) = v(f ). As we know from the previous section, the family T F generates the natural differential structure T C on T M . The base D of U T F is given by: Let (X, U), and (Y, V) be uniform spaces.

In other words for every
It is easy to prove that: (i) any uniform mapping f : (X, U) → (Y, V) is continuous with respect to topologies τ U and τ V ; (ii) a superposition of uniform mappings is a uniform mapping.
We can give criteria of the uniformity:  (c) For every pseudometric ̺ in Y uniform with respect to V, a pseudometric σ in X given by the formula x, y ∈ X, is uniform with respect to the uniform structure U.
For the proof see [6].
A mapping f , that is uniform with respect to uniform structures U and V could not be uniform with respect to another uniform structures U and V defined on X and Y respectively even if topologies τ U = τ U and τ V = τ V .
Let V be a standard uniform structure on R. Then the map f is uniform with respect to U F2 and V, but it is not uniform with respect to U F1 and V. In fact, if V = {(x, y) ∈ R : |x − y| < ε} ∈ V, then does not exists U ∈ U F1 such that   [5], Definition 2.2).
If we define the ball K(x, V ) as a set: If F ⊂ X and V ∈ D we define the V-neighbourhood of F as a set DEFINITION 3.5 A nonempty family F of subsets of a set X is said to be a filter on X if: DEFINITION 3.6 A filtering base on X is a nonempty family B of subsets of X such that If B is a filtering base on X then is a filter on X. It is called the filter defined by B and B is called the base of F . PROPOSITION 3.1 If {F i } i∈I is the family of filters on the set X then the intersection i∈I F i is a filter on X.
Proof. It is obvious that i∈I F i fulfils (F3). Suppose now that F ∈ i∈I F i . Then Let us consider now two arbitrary elements F, G ∈ i∈I F i . For any i ∈ I we have F, G ∈ F i and therefore (by (F2)) F ∩ G ∈ F i . Hence F ∩ G ∈ i∈I F i . It means that i∈I F i fulfils (F2). DEFINITION 3.7 Let X be a topological space. We say that a filter F on X is convergent to x ∈ X (F → x) if for any neighbourhood U of x there exists F ∈ F such that F ⊂ U (i.e. U ∈ F ).

PROPOSITION 3.2 If X is a topological space and for any
Proof. For any neighbourhood U of x and any i ∈ I we have U ∈ F i . Hence U ∈ i∈I F i . It means that i∈I F i → x.
We say that two Cauchy filters F 1 and F 2 are in the relation R if PROPOSITION 3.3 Two filters F 1 and F 2 on the uniform space (X, U) are in the relation R iff F 1 , F 2 and F 1 ∩ F 2 are Cauchy filters on X.
Proof. (⇒) Suppose F 1 and F 2 to be in the relation R and fix V ∈ U. Let W ∈ U be such that 4W ⊂ V . There exist F 1 ∈ F 1 and F 2 ∈ F 2 such that On the other hand, for any y 1 , y 2 ∈ K(F 1 , W ) there are x 1 , x 2 ∈ F 1 such that (y 1 , x 1 ), (y 2 , x 2 ) ∈ W . For an arbitrary chosen z ∈ F 2 we have (z, x 1 ), (z, x 2 ) ∈ W . Hence (y 1 , y 2 ) ∈ 4W ⊂ V . I means that K(F 1 , W ) × K(F 1 , W ) ⊂ V . Since K(F 1 , W ) is an element of F 1 , F 2 and F 1 ∩ F 2 we obtain all this filters to be Cauchy filters.
(⇐) Suppose F 1 ∩ F 2 to be Cauchy filter on X. Fix V ∈ U, choose F ∈ F 1 ∩ F 2 such that F × F ⊂ V and put F 1 := F 2 := F . Then F 1 ∈ F 1 , F 2 ∈ F 2 and PROPOSITION 3.4 The relation R described in Definition 3.8 is an equivalence relation on the set CF (X) of all Cauchy filters on the uniform space (X, U).
Proof. It is obvious that for any Cauchy filters F 1 and F 2 on X we have F 1 RF 1 , and if F 1 RF 2 then F 2 RF 1 . Suppose now F 3 to be such a Cauchy filter on X that F 1 RF 2 and F 2 RF 3 . Fix V ∈ U and choose W ∈ U such that 2W ⊂ V . There exist For any Cauchy filter F on X we denote by [F ] the equivalence class of F with respect to the equivalence relation R given in Definition 3.8.

PROPOSITION 3.5 If {F i } i∈I is a family of Cauchy filters on an uniform space X contained in an equivalence class [F ] then
and therefore F i ⊂ K(F, W ). Consequently K(F, W ) ∈ F i for any i ∈ I. Then K(F, W ) ∈ i∈I F i and moreover K(F, W ) × K(F, W ) ⊂ V . It means that i∈I F i is a Cauchy filter on X. Since K(F, W ) ∈ F (F ⊂ K(F, W )) we obtain that i∈I F i is equivalent to F .   For the proof see [1], [6] or [7].
It is well known that the uniform space of reals (R, U) with the standard uniform structure U = U {id R } defined by the one element family of functions {id R } (or by the standard metric) is complete. We have also more general PROPOSITION 3.6 For any set I the uniform space (R I , U G ), where G = {pr i } i∈I is the set of all natural projections pr i : R I → R, for any i ∈ I, is complete. Proof. If F is a Cauchy filter on R I then for any i ∈ I the set pr i (F ) = {pr i (F ) : F ∈ F } is a filtering base of some Cauchy filter on R. Then the Cauchy filter corresponding to pr i (F ) converges to some y i ∈ R. Putting f (i) := y i , i ∈ I we obtain function f ∈ R I such that F → f . Any uniform space can be treated as a uniform subspace of some complete uniform space. We have the following THEOREM 3.3 For each uniform space (X, U): (i) there exists a complete uniform space ( X, U ) and a set A ⊂ X dense in X (with respect to the topology τ U ) such that (X, U) is uniformly homeomorphic to (A, U A ); (ii) if the complete uniform spaces ( X 1 , U 1 ) and ( X 2 , U 2 ) satisfies condition of the point (i) then they are uniformly homeomorphic.
For the details of the proof see [1] or [7]. Here we only want to describe the construction of ( X, U ).
Let X be the set of all minimal Cauchy filters in X. For every V ∈ U we denote by V the set of all pairs (F 1 , F 2 ) of minimal Cauchy's filters, which have a common element being a small set of rank V. We define a family U of subsets of set X × X as the smallest uniform structure on X containing all sets from the family { V : V ∈ U}.
Let N be a set, M ⊆ N , M = Ø, C be a differential structure on M . DEFINITION 3.11. The differential structure D on N is an extension of the differential structure C from the set M to the set N if C = D M (if we get the structure C by localization of the structure D to M ).
For the sets N, M and the differential structure C on M we can construct many different extensions of the structure M to N . EXAMPLE 3.4. If for each function f ∈ C we assign f 0 ∈ R N such that f 0|M = f and f 0|N \M ≡ 0, then the differential structure generated on N by the family of functions {f 0 } f ∈C is an extension of C from M to N . Similarly, if for each function f ∈ C we assign the family F f := {g ∈ R N : g |M = f }, then the differential structure on N generated the family of functions F := f ∈C F f is an extension of C from M to N . If the set N \M contain at least two elements, then the differential structures generated by the families {f 0 } f ∈C and F are different. DEFINITION 3.12. If τ is a topology on the set N , then the extension D of the differential structure C from M to N is continuous with respect to τ if each function f ∈ D is continuous with respect to τ (τ D ⊂ τ ).
If on the set N there exists a continuous with respect to some topology τ extension of the differential structure C from the set M ⊂ N , then the structure C is said to be extendable from the set M to the topological space (N, τ ). EXAMPLE 3.5. The differential structure C ∞ (R) Q is extendable from the set of rationales to the set of reals. The continuous extensions are e.g. C ∞ (R) and the structure D generated on R by the family of the functions Proof. Let φ G be the generator embedding of the differential space (M, C) into the Cartesian space (R G , C ∞ (R G )) defined by G. Then the closure φ G (M ) is a complete subspace of the complete uniform space R G (see Theorem 3.2 and Proposition 3.6). We know that R G : (M, C) → (φ G (M ), C ∞ (R G ) φG (M) ) is a diffeomorphism and pr g • φ G = g for any g ∈ G. Moreover φ G (M ) is dense in φ G (M ). Then identifying any g ∈ G with π g |φG(M) , C with C ∞ (R G ) φG (M) and putting M := φ G (M ) we obtain thatg should be identify with pr g |φG (M) and D = C ∞ (R G ) φG (M) . We have also τ D = τ M = τ UG , where τ M is the topology of M treated as a topological subspace of R G . DEFINITION 3.12. The differential space ( M , D) constructed in Proposition 3.7 will be called the differential completion of the differential space (M, C) defined by the family of generators G. The set M will be denoted by compl G M and the differential structure D will be denoted by compl G C.

The maximal differential completion
Let us consider two families G and H of generators of a differential structure C on a set M = ∅. If G ⊂ H then for uniform structures U G and U H we have: U G ⊂ U H . Consequently any Cauchy filter with respect to U H is a Cauchy filter with respect to U G . In particular any minimal Cauchy filter with respect to U H is a Cauchy (but not necessarily minimal Cauchy) filter with respect to U G . This defines the natural map ι GH : compl H M → compl G M as follows: for any F ∈ compl H M the value ι GH (F ) ∈ compl G M is the minimal Cauchy filter equivalent to F with respect to the uniform structure U G . PROPOSITION 4.1 For any two families G and H of generators of a differential structure C on a set M = ∅ such that G ⊂ H the map ι GH : compl H M → compl G M defined above is smooth with respect to differential structures compl H C and compl G C.
Proof. For smoothness of ι GH it is enough to prove that for any g ∈ G the functioñ g G • ι GH ∈ compl H C, whereg G denotes the continuous extension of g onto compl G M . Since any g ∈ G is an element of H we have for each Cauchy filter whereg H denotes the continuous extension of g onto compl H M . Henceg G • ι GH = g H ∈ compl H C. such that ι D|M = id M ; (ii) for any function g ∈ C there exists uniquely defined extensiong ∈ compl C C.
In the set of all differential completions of the space (M, C) we can define an ordering relation such that: where G and H are families of generators of the structure C.
The above theorem says that compl C M is the maximal with the respect to the order completion of M which can be constructed using a set of generators of the differential structure C while compl C C is the maximal continuous extension of C from M to compl C M . DEFINITION 4.1. We will call the differential space (compl C M, compl C C) the maximal differential completion of the differential space (M, C).
Let us consider the situation when for some family of generators G the uniform space (M, U G ) is complete. and In particular compl C M = M and compl C C = C.
Proof. Any element of compl H M is represented by some filter F in M which is a Cauchy filter with respect to U H . Then F is a Cauchy filter with respect to U G and therefore F can be identify with its limit in M. Then compl H M ⊂ M On the other hand for any element p ∈ M the filter F p of all neighbourhoods of p is a Cauchy filter with respect to U H . Hence we can write M ⊂ compl H M .
The equality (4) is an immediate consequence of the definition of compl H C and the equality (3).
Let us consider the case when G = C. Proof. Suppose that (M, U C ) is not complete. Then there exists x ∈ compl C M \M . Let χ : C → R be a functional given by the formula whereg is the continuous extension of g from M to compl C M . This functional is an element of the spectrum of the algebra C but it is not an evaluation functional on M (the algebra compl C C separates points of the space compl C M ). Then C does not posses the spectral property. It is a contradiction with Theorem 1 and Corollary 6 from the work [3] (see also Theorem 2.3 (Twierdzenie 2.3) and Corollary 2.6 (Wniosek 2.6) from [2]). COROLLARY 4.1 Let X be a topological Hausdorff space. If the topology of X is given by a countable family G of real-valued functions as the weakest topology on X with respect to which all elements of G are continuous then there exists an uniform structure U on X such that the uniform space (X, U) is complete and the topology τ U coincides with the initial topology on X.
For any Hausdorff differential space we consider the generators embedding of that space using the family of generators described in Theorem 5.1 (it takes values in the cube [−1, 1] I ). So we have the embedding of the differential space into the compact space and we close the image. Hence we get the compact set M G .
If we mark J := [−1, 1], then M G ⊂ J I . On J I there exists the natural differential structure C ∞ (J I ) = C ∞ (R I )| J I generated by the family of projections {pr i|J } i∈I , where pr i : R I → R is the projection onto i-th coordinate. By localization of that structure to the set M G we get the differential space (M G , C ∞ (J I ) M G ) which is a differential subspace of the space (J I , C ∞ (J I )). We call that differential space the (differential) compactification of the differential space (M, C) by the family of generators G and we denote it by (compt G M, compt G C). We see that compt G M = compl G M and compt G C = compl G C. So for the compactification of the differential space we have analogous theorems like for the completion. Let consider the differential space (M, C) and the family C 0 of all smooth functions on that space which takes values in [−1, 1]. Using the procedure of the compactification, described earlier, we get the differential space that we mark by (compt C M, compt C C) it means compt C M := compt C0 M , compt C C := compt C0 C.
DEFINITION 5.1 The differential space (compt C M, compt C C) is called the maximal differential compactification of the space (M, C) Let us assume that the topological space (M, τ C ) is compact. Than all the functions from C are limited and by the normalization of each function α ∈ C \ {0} according the formula: N α(p) = 1 sup q∈M |α(q)| α(p), p ∈ M we get the family N C = {N α : α ∈ C} of the generators of the structure C. Then the generators embedding given by N C converts diffeomorphically (M, C) onto the compact subspace of (J N C , C ∞ (J N C )). Similarly, if G is any family of generators of the structure C, then N G = {N α : α ∈ G} is the family of generators of C too, and proper generators embedding is the diffeomorphism of (M, C) onto a compact subspace of (J N G , C ∞ (J N G )). We have where the equality is the identification of the diffeomorphic spaces and structures.