A Formalization of the Concept of a Numeral System

We consider finite and unconditionally convergent infinite expansions of elements of a given topological monoid G in some base B c G as words of the alphabet B, identify insignificantly different words and define a multiplication and a topology on the set of classes of these words. Classical numeral systems are particular cases of this construction. Then we study algebraic and topological properties of the obtained monoid and, for some cases, find conditions under which it is canonically topologically isomorphic to the initial one.


Introduction
Different numeral systems are used for calculations with real and complex numbers and begin being used for other topological algebraic structures. In particular, a wide survey of positional numeral systems in R can be found in [10]. The numeral system in C with the radix 2i and the digits 0, 1, 2, 3 (s.c. quarter-imaginary one) is also proposed there. Other numeral systems in C are considered in [5] and [8]. Some numeral systems in R[x]/(x n + 1) with n ≥ 2 are studied in [9]. Calculations in these structures are used in physics.
A. D. Markovskii, an author of many algorithms of fast calculations (see [11], [12]), widely uses non-standard representations of reals. The base B = {x n /n!} n∈N 0 in G = R[x]/(L) with L = a 0 x k + . . . + a k can be used for the approximate solving of the linear differential equation a 0 x (k) + . . . + a k x = 0 since co-ordinates of the element exp(xt) = ∞ 0 t n x n /n! in the base 1, . . . , x k−1 of G depending on the real parameter t form a fundamental system of its solutions with the unit Cauchy matrix at the point t = 0.
Generally, a numeral system in a topological monoid G assigns to its elements their expansions in some base B ⊂ G, i.e. represents them as words of the alphabet B. Endowing the set of these words with algebraic and topological structures simulating the corresponding structures of G, we can reduce approximate calculations with these elements to transformations of words. Moreover, as in the case of numbers, such representations can sometimes be useful for proofs of properties of this monoid.
The main purpose of this paper is to find constructions of these structures for arbitrary B and G. In paper [1], it was done for abelian topological groups. This paper also contains a lot of examples, but there are no theorems proved in this paper there. In papers [2] and [3], the results of [1] were partially transferred onto n-semigroups. However, it became clear now that topological monoids are the natural domain of application of these ideas.
First, we introduce a topology on the free monoid W generated by the set of expansions in the base B and prove some its properties, in particular, a criterion of compactness of its subsets. Then we identify expansions which can be obtained from each other by means of simple transformations. This equivalence is said to be algebraic. It is a congruence, and we obtain the quotient monoid W .
In order to reproduce the main topological properties of G on W , we define its topology as the quotient one in the category where G is studied. Three cases occur in this paper which are not the only possible or mutually exclusive ones. These are the case T) for the category of topological monoids, the case CR) where the underlying space of G is assumed to be T 3 1 2 , and the case U) for uniform monoids.
This topology on W can be non-T 0 . Therefore, we identify its topologically not distinguishable elements and obtain the quotient monoid W. For example, for the decimal numeral system in (R + 0 , +), the expansions 1, 00 . . . and 0, 99 . . . are topologically equivalent.
Any identity which is true for one of the monoids G, W and W, is also true for the others.
Then we define the concept of a numeral system which remained informal until now: a B-numeral system is a section of the canonical surmorphism π: W → G assigning to each element of the linear hull B its class of finite expansions. We prove our main theorem which was announced in [2]: In the cases CR) and U), the following statements are equivalent: i) There exists a continuous (in the case CR)) or uniformly continuous (in the case U)) B-numeral system; ii) The map π is an isomorphism of topological (respectively, uniform) monoids.
The third statement of this theorem links the existence of a continuous B-numeral system with the possibility of an extension onto G of a continuous map given on B .
In the last section, we show that the standard numeral systems in (R + 0 , +) are particular cases of the above construction.
1 The construction and properties of the monoid of expansions A) For a given Hausdorff topological monoid G, denote its multiplication by (we write a b instead of (a, b)), its topology by τ and its identity by 1. Let B be its fixed subset. Definition 1.1. A permissible simple (or 1-tuple) word of the alphabet B is any sequence B = {b i } i∈N of elements of B satisfying the following condition: For each neighborhood O of 1, there exists a natural n such that the values of the product i∈I b i belongs to O for any finite or infinite strictly increasing sequence of natural numbers I = {i 1 , i 2 , . . .} with i 1 > n.
For a series in a linear normed space, to meet this requirement means to be unconditionally convergent.
The meaningπ(B) of a given permissible simple word B is the product of all its letters:π(B) = i∈N b i . This product converges by the previous definition. Definition 1.2. The subset B is called a computing base if: i) 1 ∈ B; ii) Each element of G is the meaning of a permissible simple word.
The first condition has a rather technical character. It allows us not to consider separately words of a finite length. A given permissible simple word is said to be finite if its letters are eventually equal to 1.
In the following, we assume that the considered subset B is a computing base and all considered words of this alphabet are permissible. In particular, we will omit the word "permissible". It is evident that the linear hull B of such a subset B is dense in G.
Let W 1 denote the set of all simple words, and ( W, •) be the free monoid generated by this set. It is called the monoid of expansions in the alphabet B. Its elements are finite ordered collections of simple words. If B = (B 1 , . . . , B k ), k ∈ N 0 , is such a collection, then we call the words B 1 , . . . , B k the components of B and B a k-tuple word of the alphabet B. A multituple word is said to be finite if all its components are finite.The submonoid of W consisting of all finite words is denoted by FW. If 1 is an isolated point in G, then all words are finite.
Let nowN be the one-point compactification of N, i.e. the setN = N∪{∞} endowed with the order topology: each n ∈ N is an isolated point inN, and the collection of intervals of the form (n 0 , →) = {n ∈N : n > n 0 }, n 0 ∈ N, is a base of neighborhoods of the element ∞. Assign to each natural k a copŷ N k of the spaceN, consider the product k∈NNk with the product topology, and denote byÎ the subspace of this product consisting of all elements with increasing sequences of coordinates where finite coordinates are increasing strictly. This subspace is compact.
With each 1-tuple word B = {b i }, one can associate a map δ B :Î → G. We also denote it as B if it does not lead to confusion. For I = I z = (∞, . . .), this map is defined by the formula δ B (I) = B(I) = b i where i runs finite coordinates of this element I ∈Î. Moreover, set δ B (I z ) = 1.
Let I ⊂Î be the dense subspace consisting of elements with a finite number of finite coordinates (including I z ). We may consider elements of I as finite subsets of N. Order I by inclusion. Then the function δ B | I is a net in G. We call it the net of approximations corresponding to B. Its limit in G is the meaningπ(B) of this word, andπ(B) = B(I u ) where I u = (1, 2, . . .).
Let now B = (B 1 , . . . , B k ) be any k-tuple word. Order the set I k by the relation ≤ defined by (I 1 , . . . , I k ) ≤ (I 1 , . . . , I k ) iff I s ⊂ I s for all s = 1, k.
With each its element I = (I 1 , . . . , I k ), one can associate the element B(I) = B 1 (I 1 ) . . . B k (I k ) ∈ G. The resulting net is called the net of approximations corresponding to this word B. If g 1 , . . . , g k are the meanings of its components B 1 , . . . , B k , respectively, then the limit of this net is g = g 1 . . . g k . We call it the meaning of this word B. This word is called an expansion of g in the alphabet B. If a simple word B = {b i } is an expansion of g, then we write sometimes g = b i or, in the case of additive denotations, g = b i . The net of approximations of the empty word is, by definition, the sequence 1, 1, . . . and its limit is equal to 1. Thus, the following statement is true. Proposition 1.3. Assigning to each word its meaning, we obtain a surmorphism of monoidsπ : W → G, andπ( FW) = B . Remark. The existing numeral systems in commutative monoids (in (R, +), in (C, +) etc.) often use expansions of the form q i b i with coefficients (digits) q i from some set of their admissible values. In order to reduce this situation to the above one, it suffices to consider all elements of G of the form q i b i as separate elements of B.
B) Now, we introduce a topology on the set W. First, we consider the subset W 1 of simple words. Proposition 1.4. The map δ B is continuous.
If all coordinates of I are finite, let k be a natural such that b i 1 . . . b i k ∈ U and i k+1 > n 0 where n 0 is defined by the neighborhood O as in Definition 1.1. If I has only a finite number of finite coordinates, then k + 1 is the number of its first infinite coordinate. Let V denote the neighborhood of I consisting of all points whose first k coordinates are equal to i 1 , . . . , i k and the remaining coordinates are greater than n 0 . Then δ B (V ) ⊂ U . Consider now the compact-open topology on GÎ, and letτ be its preimage on the set W 1 under the map δ. With this topology on W 1 , the map δ is an embedding, and we will sometimes identify W 1 with its image. The next corollaries follow from Proposition 1.4 and well-known theorems about spaces of maps (see [6], Theorems 3.4.13, 3.4.15, 2.1.6 and 2.6.4). Then the uniformity {Û} on W 1 generates the topologyτ . This uniformity is said to be induced by {U}.
Extend the topologyτ (the uniformity {Û}) to the set W k of all k-tuple words as the k-th degree of the topologyτ (of the uniformity {Û}) on W 1 , and to W as to the sum ⊕ k W k . The multiplication • in W is continuous (uniformly continuous) in this topology (respectively, in this uniformity). Proposition 1.8. The homomorphismπ is continuous with respect to the topologiesτ and τ . If the monoid (G, , {U}) is uniform, thenπ is uniformly continuous with respect to the uniformities {Û} and {U}. Proof. The statement follows from properties of the mapπ| W 1 .
Remark. There exists another method to introduce a topology on W k . The formula δ B ((I 1 , . . . , I k )) = δ B 1 (I 1 ) . . . δ B k (I k ) where B = (B 1 , . . . , B k ) ∈ W k , defines the map δ B :Î k → G. The function B → δ B is an embedding of the set W k into the set of continuous maps GÎ k . Then the compact-open topology on this set defines a topology on W k . One can prove that if G is a T 3 1 2 space, then this topology on W k coincides with the above one. The main part of this proof is similar to the proof of Proposition 3 from [2].  If ω(G) ≥ ℵ 0 , then ω( W) ≤ ω(G).
(ii) Let χ(G), χ( W) denote characters of the spaces G and W. If Proof. (i) The weight of the spaceÎ is equal ℵ 0 , and this space is compact. Therefore, the weight of GÎ with the compact-open topology does not exceed ω(G) (see Theorem 3.4.16 from [6]). Hence, W 1 , W k = ( W 1 ) k and W = ⊕ W k have this property, too, and the first statement is proved.
(ii) It suffices to consider the space W 1 . For each 1-tuple word B, we will construct a base of this space at the point B whose cardinality does not exceed χ(G). For that, we consider elements of G of the form B(I) with I ∈ I. We assume that O α ⊂ O and find the number n 0 = n Oα (B). For each I ∈ H, denote by ν(I) the element fromÎ whose finite coordinates are the coordinates of I which do not exceed n 0 . Then the inclusion B(I) ∈ B(ν(I)) O α is true by the definition of the function n O , and so the inclusion B(ν(I)) ∈ U is true, too. The set {ν(I) : I ∈ H} is finite since it can only contain I z and elements of I with finite coordinates not exceeding n 0 . Denote elements of this set by I 1 , . . . , I s and consider neighborhoods U 1 , . . . , U s of the elements B(I 1 ), . . . , B(I s ), respectively, belonging to the selected above bases and such that U 1 , . . . , U s ⊂ U . If an element B belongs to the intersection of the sets V U k (I k ) with k = 1, s and the set W α , then B (I) ∈ B (ν(I)) O α ⊂ U O α ⊂ U for any I ∈ H. That completes the proof. Now, we are going to establish a criterion of the compactness for subspaces of W 1 and need some more terminology and denotations for that. Consider W 1 as a subspace of GÎ. For any I fromÎ, denote by γ I the map W 1 → G defined by the formula γ I (B) = B(I), and by I k , k ∈ N , the element (k, ∞, . . .) ∈Î. We say that a subspace F ⊂ W 1 is a subspace of the equiconvergence if, for any neighborhood O of 1, the function n O is bounded on F. Theorem 1.11. Let G be a T 3 -space. A subspace F ⊂ W 1 is compact if and only if the following conditions are satisfied: (i) F is closed; (ii) F is a subspace of the equiconvergence; (iii) For any k ∈ N , the subspace γ I k (F) ⊂ G consisting of k-th coordinates of elements of F is closed; (iv) The closure of the subspace Ω(F ×Î) ⊂ G is compact. Proof. If F is compact, then its equiconvergence property follows from Lemma 1.9. The subspace Ω(F ×Î) and all subspaces of the form γ I (F) with I ∈Î are compact since the maps Ω and γ I are continuous.
The proof of the sufficiency of conditions (i)-(iv) is based on the Ascoli Theorem (see, for example, [6], Theorem 3.4.20). We need to show that F is closed in GÎ and is an evenly continuous set of maps ofÎ into G.
Suppose a map ϕ 0 ∈ GÎ is a limit point of F in GÎ. We will construct a 1-tuple word B so that δ B = ϕ 0 . First, let U be an arbitrary neighborhood of the point ϕ 0 (I k ) in G. The set (I k |U ) = {ϕ ∈ GÎ : ϕ(I k ) ∈ U } is a neighborhood of ϕ 0 and, therefore, has a non-empty intersection with F. Thus, any neighborhood of the point ϕ 0 (I k ) contains elements from γ I k (F) and ϕ 0 (I k ) ∈ γ I k (F) ⊂ B since γ I k (F) is closed in G. Denote ϕ 0 (I k ) = b k , and let B = {b k } k∈N . We will prove that B is the required word.
Show that F is evenly continuous. For any g ∈ G, its arbitrary neighborhood U and an arbitrary I = (i 1 , i 2 , . . .) ∈Î, we need to find neighborhoods If all coordinates of I are finite, then we put K 1 = Ω(F ×Î) \ U and K 2 = γ I (F) ∩ V . These sets are compact, and their intersection is empty.
Indeed, let {O α } be a base of neighborhoods of 1 ordered by inclusion and suppose that, for any neighborhood O α , there are x α ∈ K 1 , y α ∈ K 2 and u α ∈ O α such that x α u α = y α . If (x, y) is a limit point of the net {x α , y α } in the compact set K 1 × K 2 , then x = y which is impossible.
Put now O = O ∩ O if all coordinates of I are finite, and O = O otherwise. In the first case, let k + 1 be the number of the first coordinate of the element I which is greater than n O = max B∈F n O (B), and, in the second case, it is the number of its first infinite coordinate. Denote by W the neighborhood of I consisting of all points whose first k coordinates are equal to the corresponding coordinates of I and the remaining coordinates are greater than n O . In the first case, denote I k = (i 1 , . . . , i k , ∞, . . .). If In this section, we consider the operations of the choice of a subsequence and of the shortening of simple words. We use them in the following.
First, we introduce a structure of a topological monoid onÎ and consider its action on W 1 . For It is not difficult to prove the next statement. Lemma 1.12. (i) The setÎ endowed with the operation * and the above topology is a topological monoid. The element I u = (1, 2, 3, . . .) is its identity and the element I z = (∞, ∞, . . .) is its zero.
(ii) Considering elements fromÎ as subsets ofN, we orderÎ by inclusion: To prove the second statement, consider an entourage U of the given uniformity on G and the corresponding entourageÛ of the uniformity on W 1 . Let B , B be 1-tuple words such that (B , B ) ∈Û, i.e. such that (δ B (K), δ B (K)) ∈ U for all K ∈Î. Then (δ B I (K), δ B I (K)) = (δ B (I * K), δ B (I * K)) ∈ U for any K ∈Î and (B I, B I) ∈Û. Corollary 1.14. Let {J(n)} n∈N be a cofinal sequence of elements of I. Then, for any simple word B, the sequence {BJ(n)} consists of finite words and converges to B. Proof. Since lim J(n) = I u , it suffices to apply the previous theorem. Corollary 1.15. The set FW of finite words is dense in W. Corollary 1.16. For each n ∈ N, denote I(n) = (1, . . . , n, ∞, . . .). The map λ( , I(n)) is a retraction onto the subspace consisting of all words of the form (b 1 , . . . , b n , 1, . . .). If the topology of G is T 3 1 2 , then this retraction is uniformly continuous with respect to the corresponding uniformity on W 1 .

The monoids of classes of algebraic and of topological equivalence
A) Here, we define the relation of the algebraic equivalence on W. First, denote by R a the following binary relation on W: for any B , B ∈ W, B R a B means that the nets of approximations of B and B have a common cofinal subsequence.
B R a B arises as a consequence of finite equalities connecting segments of B and B in this monoid. In particular, for the decimal numeral system in G = (R + 0 , +), pairs of expansions possessing this property correspond to different interpretations of the same standard representation of a real number as the sum of a series of the form n k 10 k with n k ∈ N 0 : 0, 124 = 12 · 10 −2 + 40 · 10 −4 = . . .. But (0, 9, 1) / ∈ R a . Definition 2.1. The transitive closure of the relation R a is called the relation of the algebraic equivalence.
We denote this equivalence relation by ∼ a , the quotient set W/ ∼ a by W , the natural map W → W by α, and α( FW) by FW . Proposition 2.2. Any two algebraically equivalent words have equal meanings. Any two finite words with equal meanings are algebraically equivalent. Proof. The first statement is evident. To prove the second one, let B be a ktuple finite word with the meaning g. Then B(I(n) k ) = g for all large enough n, whence the statement follows. Corollary 2.3. The mapπ: W → G induces a map π : W → G; the restriction π | F W is injective, and its image is equal to B . Proposition 2.4. The relation of the algebraic equivalence is a congruence.
s for all i = 1, r s − 1 and s = 1, 2. Then, according to the above argument, every pair of neighboring members of the sequence Denote by the multiplication of the quotient monoid structure on W . Corollary 2.5. The map π is a homomorphism of (W , ) onto G.
B) Here, we introduce a topology on the set W . It depends on the type of functions on G whose values we intend to compute by means of the considered numeral system. Three cases occur in this paper which are not the only possible or mutually exclusive ones. In the case U, (G, , {U}) is assumed to be a uniform monoid, and we want to study only uniformly continuous functions (with not necessarily T 0 ranges). In the case CR, we assume that the underlying space of G is a T 3 1 2 -space, and we will study its continuous maps into completely regular (not necessarily T 0 ) spaces. In particular, G can be a topological group. In the case T, this underlying space is studied as an object of the category TOP of topological spaces and their continuous maps.
It was proved above that, in each of considered cases, the space W is an object and the mapπ is a morphism of the corresponding category (i.e. UNIF for the case U, COMPL.REG. for the case CR and TOP for the case T).
In each case, we define the topology on W so that α is a quotient in this category map. It guarantees that the map π is its morphism and, in particular, makes W a uniform (a uniformisable) space in the case U (respectively, CR) when W possesses this property.
Denote by W T (W CR , W U ) the set W endowed with the topology corresponding to the case T (respectively, CR and U). W U is a uniform (not necessary T 0 ) space, and W CR is completely regular.
The chosen topology on W T is equal to the least upper bound of the pre-images of the topologies on all possible topological spaces M for which there exist a map ϕ and a continuous mapφ such that the diagram To obtain the topology on W CR (W U ), it is necessary to assume in addition that M is a completely regular space (respectively, M is a uniform space andφ is a uniformly continuous map).
It is evident that in the case, when the monoid G is uniform, there exist natural bijective continuous maps W T → W CR → W U , and in the case, when only the topology τ is uniform, there exists a natural bijective continuous map W T → W CR .
C) Consider now the commutative case. Proposition 2.6. If the multiplication in G is commutative, then there exists a continuous map η: W → W 1 such that η(B) ∼ a B for any word B ∈ W and, moreover, η(B) is a finite word iff B is a finite one. If is uniformly continuous, then this map η is uniformly continuous, too.
Each term of the net of approximations of B is equal to the product of appropriate terms of the nets of approximations of the components of B. Therefore, the map η is uniformly continuous if is uniformly continuous.
Verify now that η is continuous if is only continuous. Let (K|U ) be a containing B element of the subbasis of the topologyτ on W 1 defined in section 1.B). Here, K is a compact subset ofÎ, U is an open subset of G, and Let n be a natural such that n ≥ n i for i = 1, k and kn ≥ n 0 . For each I ∈ K, denote by ν(I) the element from I whose finite coordinates are equal to those coordinates of I which do not exceed kn. The element B (ν(I)) belongs to U since B (I) ∈ (B (ν(I)) O) ∩ B (K). There is only a finite number of these elements, and each of them is equal to a product of elements of the form b ij where i = 1, k, j = 1, n. Therefore, there exist neighborhoods V ij of the elements b ij such that if we replace each multiplier of the form b ij of each product of the form B (ν(I)) with any elementb ij from V ij , then all these productsB (ν(I)) will belong to U as before.
It is a neighborhood of the word B. For each its elementB and each element I ∈ K, we have Denote by α 1 the restriction α| W 1 . Proposition 2.7. If the multiplication is commutative, then, in each of the considered cases, the map α 1 is a quotient in the corresponding category map onto the corresponding space W T (or W CR , W U ). Proof. According to Proposition 2.6, the quotient map α is the composition of the morphisms W Remark. Propositions 2.6 and 2.7 show that the method by means of which the algebraic and the topological structures on W U are defined in this paper, gives the same results as the method of the paper [1] does. D) Consider now the question of the continuity of the operation . Proposition 2.8. In the case U, the operation is uniformly continuous. In two other considered cases, it is continuous in each argument separately. Thus, in the case U, (W U , ) is a uniform monoid on a not necessarily T 0 uniform space. In the case CR, (W CR , ) is a semitopological monoid on a completely regular underlying space, and in the case T, (W T , ) is a semitopological monoid on a not necessarily T 0 topological space. Proof. In the case U, the statement follows from the fact that the product of quotient in the category UNIF maps is a quotient in this category map, too (see [7]). In the remaining cases, it follows from Proposition 2.4 and the definition of a quotient map that right (left) translations in W CR and W T are continuous since they are continuous in W.
E) We define now the relation of the topological equivalence. Considering all three cases simultaneously, denote by K any of the symbols U , CR, T . Generally speaking, the topology on W K is not T 0 . Consider, in connection with it, the binary relation in the set W × W denoted by ∼ t K and called the relation of the topological equivalence of the type K.
For elements B , B ∈ W , we put B ∼ t K B if B and B are topologically indistinguishable, i.e. any open subset of W K containing one of B , B contains them both.
The relation ∼ t U is the intersection of all entourages of the uniformity on W U , and the relation ∼ t CR is the intersection of all entourages of all uniformities compatible with the topology on W CR . Each open set in W K is saturated with respect to the relation ∼ t K , and each continuous function on W K into any T 0 -space is constant on each class of ∼ t K .
Denote now by W K the quotient space W K / ∼ t K in the category corresponding to the case K and by θ K the quotient map W K → W K . They are an object and a morphism of this category. W U ( W CR ) is the Hausdorff uniform (respectively, uniformizable) space associated with the initial space W U (W CR ). W T is the Kolmogorov quotient of W T .
In the case K, we will say that words B 1 , B 2 from W are t K -a-equivalent if they have the same images by the map θ K • α : W → W K . Such words have equal meanings. For any K, this map θ K • α is quotient in the corresponding category since it is a composition of maps possessing this property.
The map θ K is open, and its restriction to any subspace of W K is a quotient in this category map, too. The restriction of the map θ K to the subspace FW K (i.e. subset FW endowed with the topology of a subspace of W K ) is bijective since the restriction of the map π K to this subspace is bijective and continuous by Corollary 2.4. We denote the image of FW K by FW K . It follows from Corollary 1.15 that it is a dense subset of W K .
Let M be a uniform space, and φ: M → W U be a map. This map is uniformly continuous if and only if the map θ U • φ is uniformly continuous. Similar statements are true in the cases CR and T.
If G is a uniform monoid, then there exist three relations of topological equivalence corresponding to the different definitions of the topology on the set W : ∼ t T , ∼ t CR and ∼ t U , and each of these relations is contained in the next one. Thus, we obtain three quotient spaces connected by continuous surjective maps W T tcr −→ W CR cru −→ W U . If only the topology τ is uniform, then there exist spaces W T and W CR connected by the map tcr.
It easily follows from Proposition 2.8. that the relation of the topological equivalence of the type K is a congruence in (W , ) for any K. We denote by • the result of the transposition of the operation onto W K .
In the case U, its uniform continuity can be proved by the same argument as in Proposition 2.8. In the remaining two cases, this operation is continuous in each argument.
The map π induces a surmorphism π K : (W K , •) → G which is continuous in the cases T and CR and is uniformly continuous in the case U.
We have proved the following statement. Proposition 2.9. (i) The set W T endowed with the defined above topology and with the multiplication • is a semitopological monoid on a T 0 -space, and the map π T is a continuous surmorphism.
(ii) The set W CR endowed with the defined above topology and with the multiplication • is a semitopological monoid on a T 3 1 2 -space, and the map π CR is a continuous surmorphism.
(iii) The set W U endowed with the defined above uniformity and with the multiplication • is a uniform monoid, and the surmorphism π U is uniformly continuous. Corollary 2.10. The monoids W, W K , W K and G are connected by the following commutative diagram of the defined above continuous (in the cases K=T and K=CR) or uniformly continuous (in the case K=U) homomorphisms: The statement follows immediately from the definitions of the mapŝ π, π and π K .
F) Here, we consider other properties of the monoids W, W K and W K . Theorem 2.11. Let G 1 , G 2 be Hausdorff topological monoids in the case K=T, or topological monoids on T 3 1 2 -spaces in the case K=CR, or uniform monoids in the case K= U and B 1 , B 2 arbitrary computing bases in these monoids, respectively. Denote by W i , W iK and W iK the monoids of B iexpansions and their classes for G i , i = 1, 2, and K=T, or CR, or U. Let ϕ be a continuous in the cases K=T and K=CR or uniformly continuous in the case K=U identity preserving homomorphism of Then there exist canonical defined continuous in the cases K=T and K=CR and uniformly continuous in the case K=U homomorphisms of monoidŝ ψ, ψ and ψ K such that the diagram

commutes.
Proof. First, consider the cases T and CR. Define a mapψ 1 of the set W 11 of simple words of the alphabet B 1 into the set W 21 of simple words of the alphabet B 2 as follows. For B = {b i } i∈N ∈ W 11 setψ 1 (B) = {ϕ(b i )} i∈N . It is an element of W 21 . If B is empty, thenψ 1 (B) is empty. Show thatψ 1 is continuous. Denote by δ 1 (respectively, δ 2 ) the map W 11 → G 1Î (respectively, W 21 → G 2Î ) defined as above in section 1.B). Then it is easy to check that ϕ(δ 1 (B)(I)) = δ 2 (ψ 1 (B))(I) for any B ∈ W 11 and I ∈Î. Consider now the subbasis of the topology on W 21 consisting of sets of the form (K|U ) = {B ∈ W 21 : δ 2 (B)(K) ⊂ U } where K ⊂Î is compact and U ⊂ G 2 is open. Thenψ −1 1 ((K|U )) = {B ∈ W 11 : δ 1 (B)(K) ⊂ ϕ −1 (U )} is open in W 11 for any K and U such as above. It implies thatψ 1 is continuous. Now, it follows from the definitions of the monoid W in section 1.A) and its topology in section 1.B) that the mapψ 1 can be uniquelly extended to a continuous homomorphism of W 1 into W 2 . We denote this extension byψ.
Prove thatψ takes algebraically equivalent elements into algebraically equivalent ones. Let B = (B 1 , . . . , B k ) be an arbitrary k-tuple word from W 1 . First, we show that the net of approximations of the wordψ(B) ∈ W 2 is the image under ϕ of the net of approximations of B. In particular, it implies that the map ϕ takes the meaning of B into the meaning of ψ(B), i.e. ϕ •π 1 =π 2 •ψ. Indeed, we have for any I = (I 1 , . . . , I k ) ∈ I k : ψ(B)(I) =ψ 1 (B 1 )(I 1 ) . . . ψ 1 (B k )(I k ) = ϕ(B 1 (I 1 )) . . . ϕ(B k (I k )) = ϕ(B 1 (I 1 )) . . . ϕ(B k (I k )) = ϕ(B(I)). For arbitrary B , B ∈ W 1 , B R a B means that the nets of approximations of these elements have a common cofinal subsequence. The image under ϕ of this subsequence is the common cofinal subsequence of the nets of approximations of the elementsψ(B ) and ψ(B ). Hence,ψ(B )R aψ (B ). (We denote equally the relations R a , ∼ a , ∼ t in the monoids of expansions and of classes of expansions corresponding to the given monoids G 1 and G 2 and their computing bases.) Therefore, B ∼ a B impliesψ(B ) ∼ aψ (B ).
Let now B and B are elements of W 1K such that B ∼ t K B , i.e. any open subset of W 1K containing one of B , B contains them both. The proved continuity of ψ implies that ψ (B ) ∼ t K ψ (B ). Therefore, there exists an only map ψ K such that ψ K • θ 1K = θ 2K • ψ . This map is continuous since θ 1K and θ 2K are quotient maps in the corresponding categories.
The equality ϕ • π 1K = π 2K • ψ K follows from ϕ •π 1 =π 2 •ψ. Consider now the case U. It suffices to prove that the mapψ is uniformly continuous. Let the topologies τ 1 on G 1 and τ 2 on G 2 are given by uniformities with bases of entourages U 1 and U 2 , respectively, andÛ be an entourage of the corresponding uniformity on W 2 . We may assume that it has the form U = {(B , B ) ∈ W 2 × W 2 : |B (I) − B (I)| < U for any I ∈Î} where U ∈ U 2 . There exists an entourage U of the considered uniformity on G 1 such that (ϕ × ϕ)(U ) ⊂ U . Denote byÛ the corresponding entourage of the considered uniformity on W 1 . Then it is not difficult to verify that (ψ ×ψ)(Û ) ⊂Û . Corollary 2.12. In the case T, denote by C 1 the category whose objects are pairs (G, B) where G is a Hausdorff topological monoid and B is a computing base in G, and whose morphisms are continuous identity preserving homomorphisms ϕ : G 1 → G 2 such that ϕ(B 1 ) ⊂ B 2 , and by C 2 the category of Hausdorff topological monoids. Then the function P (ϕ) =ψ is a functor from C 1 into C 2 . Similar statements are true for functions P (ϕ) = ψ and P (ϕ) = ψ in the case T and for all three functions in the cases CR and U. Theorem 2.13. Any identity which is true for one of the monoids G, W and W K where K is U , CR or T , is also true for the others.
Proof. Let x 1 . . . x p = y 1 . . . y q be an identity in G written in some alphabet A. First, we will prove that it is true in W . Consider some maps ϕ: It is evident that if an identity is true in W , then it is true in W K , too. Consider now an identity in W K . It is also true in the submonoid FW K . The restriction of the map π K to this submonoid is an isomorphism onto B . This is a dense submonoid of G. Therefore, the considered identity is also true in this initial monoid.
We turn now to the commutative case and prove a statement which can serve as one of bases for the construction of positional numeral systems.
Assume that B is linearly ordered so that each simple word contains the greatest in this word letter. Denote this order by ≥. A simple word B = {b 1 , b 2 , . . .} is said to be ordered if b j ≤ b i follows from j > i for any indices i, j. Theorem 2.14. In the cases K= CR and K=U, let G be commutative and B linearly ordered as above. Then any word from W is t K -a-equivalent to an ordered one. Proof. Here, we will use additive denotations: + instead of and 0 instead of 1. Then 0 ∈ B, and we observe that, when adding or deleting zeros from a certain word, we obtain an algebraically equivalent word. Therefore, we may assume with no loss of generality that the initial word B = {b 1 , b 2 , . . .} is simple (see Proposition 2.6), infinite (see Proposition 2.2) and does not contain zeros.
Then it follows from Definition 1.1 that the number of occurrences of any letter in this word B is finite. Therefore, we can place its letters in the decreasing order. First, we prove that the obtained word B = {b 1 , b 2 , . . .} is permissible. Let O be an arbitrary neighborhood of 0 and n a natural such that the sum i∈I b i converges and its value belongs to O for any finite or infinite strictly increasing sequence of natural numbers I = (i 1 , i 2 , . . .) with i 1 > n. Denote by n a natural such that b i ∈ {b 1 , . . . , b n } for any i > n , and let r n (B) be the remainder of the word B after deletion of its first n letters and r n (B ) the similar remainder of B . Then the number of occurrences in r n (B ) of any its letter is equal to the number of its occurrences in r n (B). Hence, the value of any sum of the form i∈I b i where I = (i 1 , i 2 , . . .) is a finite or infinite strictly increasing sequence of natural numbers with i 1 > n , belongs to O.
It now suffices to prove that any neighborhood of B contains a word which is algebraically equivalent to B. Indeed, it implies that any neighborhood of α(B ) in W CR (and also in W U ) contains α(B), and then they are topologically equivalent since W CR and W U are completely regular. Denote now by k m the number of the first places of the word B which are occupied by its first m different letters together with their repetitions. As above, I(k m ) is the sequence of naturals from 1 to k m which is supplemented by symbols ∞. It is an element from I, and the sequence I(k 1 ), I(k 2 ), . . . is cofinal in I. Then, by Corollary 1.14, {B I(k m )} m∈N is a sequence of simple finite words converging to B . Therefore, Each transposition of a finite number of letters of any word leads to an algebraically equivalent one. Therefore, for any m, there exists a word which is algebraically equivalent to B, begins with the first k m letters of the word B (or B I(k m )) and whose order of the remaining letters repeats their order in B. 3 The concept of a numeral system A) We can now define the concept of a numeral system.
For each element b ∈ B , denote by β K (b) the class of its finite expansions from W K . The map β K is a monomorphism with the dense image FW K . The composition π K •β K is the identity map, and the restriction π K | of the map π K to β K ( B ) is bijective and continuous in the cases T and CR and uniformly continuous in the case U. Definition 3.1. A B-numeral system of the type K where K = T , CR or U , in G is a section of the map π K : W K → G (i.e. it is a map φ: G → W K such that π K • φ = id) which coincides with the map β K on B . Remark. Elements of W are classes of generalized words of the alphabet B. A numeral system φ assigns such a class to each element of G. The topology of the subspace φ(G) and its algebraic structure have to simulate the corresponding structures of G. Naturally, the class φ(b) corresponding to an element b ∈ B has to contain the word {b, 1, 1, . . .}. Therefore, if b 1 , . . . , b n  are elements of B, then the class φ(b 1 . . . b n ) has to contain the word {b 1 , 1, 1, . . .} • . . . • {b n , 1, 1, . . .} and is equal to β(b 1 . . . b n ).
The continuity of a given numeral system permits to regulate the error of calculations. If such a numeral system exists, then it is unique since B is dense in G. Proposition 3.2. For K = T (respectively, CR, U ), if there exists a continuous (respectively, continuous, uniformly continuous) section of the map π K , then each of the mapsπ, π K , π K is quotient in the category TOP (respectively, COMPL.REG, UNIF). Proof. Let φ be this section. It is a homeomorphism of G onto the subspace Im φ of the corresponding space W K in the cases K = T and K = CR and an isomorphism of uniform spaces G and Im φ in the case K = U . The map φ • π K is a retraction of the space W K on this subspace. Hence, it is quotient in the corresponding category. The maps π K = π K • θ K andπ = π K • α are compositions of quotient maps. Therefore, they are quotient maps, too.
We will now turn to the main theorem of this paper. For that, we need to introduce some more terminology and to prove several lemmas. Let M be a topological space, R ⊂ M × M a binary relation, and A a dense subset of M. This subset is said to be dense in R if (A × A) ∩ R is dense in R. In other words, if, for any m , m ∈ M with m Rm and for any neighborhoods V of m and V of m , there exist a , a ∈ A such that a ∈ V , a ∈ V and a Ra . and let R be a binary relation in M such that its transitive closure coincides with P. If A is a dense subset of M which is also dense in R, then the restriction p| of this map p to A (considering as a map onto p(A)) is a quotient in this category map, too. Proof. Let C be an arbitrary uniform space, C its completion, and λ, µ be maps such that λ is uniformly continuous and the diagram Here, i,î are the canonical embeddings of subspaces and j a canonical map of C into its completion. If C is Hausdorff, then this map is an embedding, too. We have to prove that µ is uniformly continuous. The map j •λ can be extended to a uniformly continuous mapλ: M → C . Show that equivalent with respect to P elements of M have the same images byλ. If m Pm , then we can assume that m Rm . Therefore, there exist convergent respectively to m and m nets {a α }, {a α } consisting of elements of A and satisfying the condition a α Ra α for all α. Then a α Pa α , λ(a α ) = λ(a α ) for all α and, therefore,λ(m ) =λ(m ). Hence, there exists a mapμ: N → C extending j •µ and such thatλ =μ•p. The map p is quotient in the category UNIF, and soμ and its restriction µ are uniformly continuous. Lemma 3.4. In the case U, the restriction α| of the map α: W → W U to the subspace FW (considered as a map onto FW U ) is a quotient map in the category UNIF. Proof. The subspace of finite words FW is dense in ( W,τ ). Show that it is dense in the binary relation R a . Consider any words B , B satisfying the condition B R a B and construct sequences of finite words {B (n)}, {B (n)}, n ∈ N, convergent to B and B , respectively, such that B (n)R a B (n) for all n. Let B be a k-tuple word, B a l-tuple word and {I (n)}, {I (n)}, n ∈ N, cofinal sequences in I k and I l , respectively, such that B (I (n)) = B (I (n)) for all n. Denote by (B 1 , . . . , B k ), (B 1 , . . . , B l ) collections of components of the words B and B and by (I 1 (n), . . . , I k (n)), (I 1 (n), . . . , I l (n)) collections of coordinates of I (n) and I (n). For each n ∈ N, it is evident that the limits of the finite words B (n) = (B 1 I 1 (n), . . . , B k I k (n)) and B (n) = (B 1 I 1 (n), . . . , B l I l (n)) coincide and, moreover, B s = lim B s I s (n) and B r = lim B r I r (n) for all s = 1, k, r = 1, l by Corollary 1.14. It remains to apply the previous lemma. Lemma 3.5. In the case CR, suppose that ( W,τ ) and W CR are endowed with their fine uniformities, i.e. with the finest uniformities which are compatible with their topologies, and FW and FW CR are endowed with the uniformities of their subspaces. Then the restriction α| of the map α to FW (considered as a map onto FW CR ) is a quotient in the category UNIF map. Proof. Since α is quotient in the category COMPL.REG (see the definition of the topology on W CR in section 2.B)) and W and W CR are considered with their fine uniformities, then the map α is also quotient in the category UNIF. Now, we can apply Lemma 3.3 together with the proved above fact that FW is dense in R a . Lemma 3.6. In the case U, if the multiplication in G is commutative, and in the case CR, if is commutative and, moreover, W 1 and W CR are endowed with their fine uniformities and their subsets FW 1 (consisting of finite simple words) and FW CR are endowed with the uniformities of their subspaces, then the restriction α 1 | of the map α 1 : W 1 → W U (respectively, α 1 : W 1 → W CR ) to FW 1 (considered as a map onto FW U or, respectively, on FW CR ) is a quotient in the category UNIF map. Proof. By Proposition 2.7, α 1 is a quotient in the corresponding category map, and we can use the arguments of lemmas 3.4 and 3.5.
We use the following terminology below. Let C be a completely regular space in the case CR or a uniform space in the case U and λ: B → C be a map. We say that this map is extendible to a continuous (respectively, uniformly continuous) map of the space W ( W 1 ) (with the topology (respectively, with the uniformity) defined in section 1.B)), if there exists a map µ: W → C (µ: W 1 → C) possessing this property of the continuity and satisfying the condition µ| F W = λ •π| F W (µ| Theorem 3.7. The following statements are equivalent in the cases CR and U: (i) There exists a continuous (in the case CR) or uniformly continuous (in the case U) B-numeral system of the corresponding type; (ii) The corresponding to the considered case map π K is an isomorphism of topological (respectively, uniform) monoids (W K , •) and G; (iii) For any completely regular in the case CR or uniform in the case U space C, each map λ: B → C which is extendible to a continuous in the case CR or uniformly continuous in the case U map of the space W, is itself continuous (respectively, uniformly continuous) and can be extended to a map of G possessing the same continuity property.
Additionally, if the multiplication in G is commutative, then properties (i)-(iii) are equivalent to the next one: (iv) For any completely regular in the case CR or uniform in the case U space C, each map λ: B → C which is extendible to a continuous in the case CR or uniformly continuous in the case U map of the space W 1 , is itself continuous (respectively, uniformly continuous) and can be extended to a map of G possessing the same continuity property. Remark to statement (iii). If λ| can be extended to a continuous (respectively, uniformly continuous) map λ of G into C, then this extension is unique since B is dense in G. Moreover, λ •π is an extension of λ| on W possessing the same continuity property. Proof. Consider the case U. The implication (ii) ⇒ (i) is evident since the map π U −1 is a uniformly continuous numeral system. (ii) ⇔ (iii). First, we will show that the statement (iii) is true if and only if the restrictionπ| of the mapπ to the subspace FW is quotient in the category UNIF. Assume that (iii) is true and consider the diagram from the proof of Lemma 3.3 with M = W, A = FW, N = G, p =π. Then p(A) = B . Let λ be uniformly continuous. It can be extended to a uniformly continuous map λ: W → C , and it implies that the map j • µ can be extended to a uniformly continuous map of W. Hence, it is uniformly continuous. Therefore, µ is uniformly continuous, too, andπ| is quotient.
Conversely, letπ| be a quotient map in UNIF and µ a map which can be extended to a uniformly continuous map of W. Denote byλ such an extension. Then λ and, hence, µ are uniformly continuous. Therefore, µ can be extended to a uniformly continuous mapμ: G → C . We have now two uniformly continuous maps of W into C : j •λ andμ•π. They coincide on the dense set FW and that's why coincide everywhere. Therefore, Imμ ⊂ j(C) and the map µ is uniformly continuously extended onto G.
We show now that the statement (ii) is true if and only if the restriction π U | of the map π U to the subspace FW U of classes of finite expansions is an isomorphism of uniform spaces. The necessity is evident. To prove the sufficiency, assume that π U | is an isomorphism. Denote byW U andG the completions of the uniform spaces W U and G, respectively. Since W U and G are Hausdorff, we can consider them as subspaces ofW U andG, respectively.
The map π U can be extended to a uniformly continuous mapπ U :W U →G. The subspace B is dense inG, and β = (π U |) −1 is its uniformly continuous map intoW U . It can be extended to a uniformly continuous map g:G →W U . The map g •π U is uniformly continuous, and its restriction to FW U is an identical map. Hence, g •π U is identical, and π U is an isomorphism between uniform spaces W U and G. Moreover, π U is a homomorphism, and (ii) is true.
Prove now thatπ| is a quotient map in UNIF if and only if the map π U | is an isomorphism of uniform spaces. First, let π U | be an isomorphism. The equalityπ = π U • θ U • α impliesπ| = π U | • θ U | • α| where α| is the restriction of the map α to FW and θ U | is the restriction of the map θ U to FW U . The map α| is quotient by Lemma 3.4 (in the case CR, it will be necessary to use Lemma 3.5). So the map θ U | is quotient as well as any restriction of θ U to a subspace. Hence, the mapπ| is quotient as a composition of quotient maps.
Conversely, suppose the mapπ| is quotient in UNIF. Then π U | is quotient, too. However, it is also bijective, since the restriction π | of the map π to FW is bijective by Corollary 2.3. Hence, π U | is an isomorphism of uniform spaces.
(ii) ⇔ (iv) if the multiplication in G is commutative. Similarly to the above argument, we can prove that (iv) is true if and only if the restriction π 1 | of the mapπ to the subspace FW 1 of finite simple words is a quotient map in the category UNIF. The equalityπ 1 | = π U | • θ U | • α 1 | is true, and the map α 1 | is quotient in the category UNIF by Lemma 3.6. Now, it is easy to verify that the mapπ 1 | is quotient if and only if the map π U | is an isomorphism of uniform spaces.
(i) ⇒ (ii). Let ϕ: G → W U be a uniformly continuous B-numeral system. Its restriction to B is back to π U |. Hence, ϕ • π U is identical, and π U is an isomorphism of uniform monoids.
For the case U, the proof is complete. To obtain the proof for the case CR, endow the spaces G, W, W CR , W CR with their fine uniformities and the spaces B , FW, FW CR , FW CR with the uniformities of their subspaces. Then W CR becomes the Hausdorff uniform space associated with the uniform space W CR . Any continuous map of any uniformizable space endowed with its fine uniformity into a uniform space is uniformly continuous. Using these facts, it is possible to verify that the above argument is also true in the case CR.
The next statement supplements the proved theorem. Proposition 3.8. In the case U (CR, T), letφ be a uniformly continuous (respectively, continuous, continuous) map of W into a Hausdorff uniform (respectively, T 3 1 2 , Hausdorff) space C such thatφ(B 1 ) =φ(B 2 ) for any finite words B 1 and B 2 with a common meaning. Then there exists a morphism of the corresponding category φ : Proof. It follows from our assumption thatφ(B 1 ) =φ(B 2 ) if B 1 R a B 2 is true. Hence, the mapφ is constant on each class of algebraic equivalence, and so it generates some map φ : W K → C. The definition of the topology on W K implies that this map is a morphism of the corresponding category.
In particular, it is continuous. Therefore, it is constant on every class of topological equivalence and induces some map φ : W K → C. This is the required map. By the definition of the topology on W K , it is a morphism of the corresponding category.
We establish now the following sufficient condition of the existence of a uniformly continuous numeral system. Proposition 3.9. The map π U is an isomorphism of uniform monoids W U and G, and there exists a uniformly continuous B-number system if, for each entourage U of the given uniformity on G, there exists its entourage V such that, for each elements g , g ∈ B with (g , g ) ∈ V, there exist their finite simple expansions B , B with (B , B ) ∈Û whereÛ is the corresponding to U entourage of the uniformity on W 1 which was defined in section 1.B). Proof. Let O be an entourage of the uniformity on W U and U an entourage of the uniformity on G such that (θ Hence, the map π U | −1 = β U is uniformly continuous, and π U | is an isomorphism of uniform monoids. Now, the statement follows from the proof of the preceding theorem.

Examples
A) The first example is especially important since we are showing that standard numeral systems in (R + 0 , +) (endowed with the usual topology) can be obtained by means of the above construction. For that, we are considering the variable-length numeral system which was defined and studied in Chapter IV of "General topology" of Bourbaki (see [4]). It generalizes classical numeral systems such as the binary one etc.
In particular, it is proved in [4] that any positive number can be written as the value of a finite or infinite sum of the form n k b i k where k, n k ∈ N and the following conditions are satisfied: if this sum consists of k 0 addends (k 0 ∈ N or k 0 = ∞), then 0 ≤ i k < i k+1 for every 1 ≤ k ≤ k 0 − 1 and We call such sums standard. Each positive number has exactly one infinite standard expansion. If it has also a finite expansion, then any sufficiently distant remainder of its standard infinite expansion has the form i≥k To use denotations of the preceding sections, add the element b −1 = 0 to B. By Proposition 2.6, any permissible word of the alphabet B is algebraically equivalent to a simple one. In this section, we will often use the additive form of the recording of simple words, i.e. we will write b i instead of {b i } i∈N . For each k, if such a sum contains n k consecutive addends b i k = 0, then we will write it as n k b i k . This is a one-to-one correspondence between simple words and convergent series with addends from B. For any non-zero b ∈ B, the number of its occurrences in such a series is finite. Finite words correspond to finite sums where the infinite number of zeros is added. Simple words with the same non-zero addends are algebraically equivalent. Operating with series, we will sometimes use the term "the value of the sum" instead of "the meaning of the word". Every standard sum corresponds to a permissible word which is also said to be standard.
First of all, we prove that any simple word is algebraically equivalent to a standard one. If this word B is finite, then it can be easily transformed to a standard one with the same meaning. Suppose now that the word B is infinite. Writing it as a sum, place brackets in it in the following way. In the first brackets, we only put the first term of B. We denote its number in B by i 1 . Assume now that the k-th brackets are already placed, the last their term is the term of the considered sum with the number r k and the greatest among the numbers of elements of B included in these brackets is i k . Then (k + 1)-th brackets begin with the term with the number r k + 1 and end with the term with the number r k+1 satisfying the following conditions: where S m is the m-th partial sum of B and the symbol [ ] denotes the whole part of a number, is stationary for m ≥ r k+1 ; ii) if i k+1 is the greatest among indices of elements from B that occur in these brackets, then i k+1 > i k .
Write now the content of the k-th brackets, k ≥ 2, as This word is algebraically equivalent to the initial one and can be written as (q k + p k+1 )b i k where q 1 = 1 and (q k+1 + p k+2 )b i k+1 < b i k for k ≥ 1 since the above sequence is stationary. Now, we transform each expression ( To obtain the required standard sum such that the corresponding word is algebraically equivalent to B, we have to omit each addend n i b i with n i = 0. The unique expansion of zero is the empty one.
Finite expansions with a common sum are algebraically equivalent by Proposition 2.3. An infinite expansion cannot be algebraically equivalent to a finite one because each term of its net of approximations is less than its meaning, whereas each sufficiently distant term of the net of approximations of any finite expansion is equal to its meaning.
Algebraic structure on W is defined by the following rules: the sum of two classes of finite expansions is a class of finite expansions, the sum of two classes of infinite expansions or a class of finite expansions with a class of infinite expansions is a class of infinite expansions. It is clear that the monoid (W , ) is not cancellable, and so the statement of Theorem 2.11 is not applicable to conditional identities.
Describe now the topologies of the spaces W T , W CR and W U . By Proposition 2.7, they are quotient spaces of the space W 1 in the corresponding categories. Therefore, we will only consider simple words.
First, we will prove that if a number a has both the class of finite expansions and the class of infinite ones, then the first of them is not closed and the second one is closed in W T . Indeed, let n i b i be the standard infinite expansion of this number a. Then there exists i 0 ∈ N such that n i = b i−1 /b i − 1 for i ≥ i 0 . Any neighborhood of this expansion contains a finite word. To find it, it is enough to add unity to one of these coefficients and to put all subsequent terms equal to zero.
Consider now the subset of W 1 consisting of all simple finite expansions of this number a and all simple expansions with the sums from interval ]a, a + [ where > 0. We will show that this subset is open. It is saturated by the relation of the algebraic equivalence, hence, its image in W T is open, too. Let B be a finite expansion of a. Denote by i the greatest among indices of elements of B that occur in B, and consider the neighborhood of B in W 1 consisting of all words whose letters differ from the corresponding non-zero letters of B by less than 1/2 b i and whose meaning is less than a + . The letters of these words corresponding to non-zero letters of B coincide with them. The meanings of these words cannot be less than a. If such a meaning is equal to a, then the remaining letters of this word are equal to zero, i.e. it is finite.
Let now U ⊂ W T be a neighborhood of the class of infinite expansions of a. We will show that it contains all classes of the algebraic equivalence with the sums from some neighborhood of a. Indeed, the pre-image of U in W 1 contains the standard infinite expansion B of a and, therefore, all words B satisfying the inequality |B (I) − B(I)| < for some > 0 and any I ∈Î. The proved property implies that the restriction of the map π to the submonoid of (W T , ) consisting of classes of infinite expansions is a homeomorphism onto R + 0 . Moreover, it is a homomorphism onto (R + 0 , +). Hence, these monoids are topologically isomorphic.
Note that the first part of the above argument does not use that the expansion B is infinite. It implies that any neighborhood of the class of finite expansions of the number a contains all classes of expansions with the sums from some interval of the form ]a, a + [. Therefore, a base of the space W T is completely described. This space satisfies the separation axiom T 0 but does not satisfy the axiom T 1 . In particular, it implies that (W T , ) and (W T , •) are topologically isomorphic.
Prove now that π U is an isomorphism of the uniform monoids W U and (R + 0 , +). It is sufficiently to show that Proposition 3.9 can be used. Let g , g ∈ B , |g − g | < δ, and g < g . Construct such finite simple expansions B and B of g and g that |B (I) − B (I)| < δ for each I ∈Î. Let B andB be any finite simple expansions of g and g − g . Denote by B the expansion of g whose terms are alternately terms of the expansionB and zeros and by B the expansion of g whose terms are alternately terms of the expansionsB andB . One can verify that the previous inequalities are true.
It remains to consider the topology of W CR . The diagram of continuous surjective maps (see section 2.E) commutes, and W CR is a T 3 1 2 -space. Therefore, the map tcr identifies classes of finite words with corresponding classes of infinite words having the same meanings. Hence, it follows from the above properties of the maps π T and π U that π CR is an isomorphism of topological monoids, too.
Thus, a continuous (uniformly continuous) B-numeral system exists in the case CR (respectively, U) and assigns to each element of R + 0 the class of its standard expansion. It does not exist in the case T although (R + 0 , +) is topologically isomorphic to a submonoid of W T . B) We consider now the computing base B in (R; +) endowed with the usual topology, consisting of zero, of all numbers 2 −n with n ∈ N 0 and of some irrational negative a. We will prove that monoid (W CR , •) is topologically isomorphic to (R + 0 , +) ⊕ (N 0 , +), and the map π CR can be defined by the formula (x, n) → x + na where x ∈ R + 0 , n ∈ N 0 . Therefore, π CR is not an isomorphism, and for this computing base there are no continuous numeral systems of the type CR.
As above, we will only consider simple words. Let B be such a word. Denote by n(B) the number of occurrences of a in it, and consider the entourage of the diagonal of W 1 × W 1 consisting of pairs (B , B ) of words satisfying the condition |B (I) − B (I)| < |a| for all I ∈Î. It can only contain pairs with the same n. Moreover, such a pair belongs to this entourage iff these words contain all letters a on the same places. Hence, the space W 1 is the sum of its subspaces W 1 (n), n ∈ N 0 , consisting of words with a fixed n, and each of these subspaces is the sum of its subspaces consisting of words containing all letters a on the same places. Each change of places of letters a gives a homeomorphism of such subspaces taking each word to an algebraically equivalent one.
Words with different n cannot be algebraically equivalent, since all distant enough terms of their nets of approximations are different. It implies that W CR is the sum of its subspaces W CR (n) = W 1 (n)/ ∼ a endowed with the quotient in the category COMPL.REG topology.
The corresponding numbers n are summed up by addition of elements of W CR . Hence, W CR (0) is a submonoid in W CR . Now, it follows from the argument of the previous example that this submonoid consists of classes of expansions of the usual binary numeral system, and, therefore, it is topologically isomorphic to (R + 0 , +).
Let now B be a word not containing a. Placing n letters a as its initial terms, we obtain some word B . It is evident that the map B → B is a homeomorphism of the subspace W 1 (0) onto the summand S n of W 1 (n) consisting of words containing n letters a as their initial terms. Words B 1 , B 2 not containing a are algebraically equivalent if and only if the corresponding them words B 1 , B 2 possess this property.
For any n, each word from W 1 (n) is algebraically equivalent to some word B whose n(B) = n first letters are a. Therefore, the quotient spaces ⊕ n (S n / ∼ a ) = (⊕ n S n )/ ∼ a and (⊕ n W 1 (n))/ ∼ a = W CR are canonically homeomorphic. The above described translation makes each space S n / ∼ a canonically homeomorphic to S 0 / ∼ a = W CR (0) ≈ R + 0 , and this homeomorphism can be defined by the formula x → x + na since this translation adds na to the meanings of all words from S 0 = W 1 (0). Moreover, the relation of the topological equivalence is trivial, since W CR is already a T 0 -space, and therefore W CR = W CR . It completes the proof.