A URYSOHN TYPE LEMMA FOR GROUPOIDS

A URYSOHN TYPE LEMMA FOR GROUPOIDS

M ˘AD ˘ALINA ROXANA BUNECI

Abstract. Starting from the observation that through groupoids we can express in a unified way the notions of fundamental system of entourages of a uniform structure on a spaceX, respectively the system of neighborhoods of the unity of a topological group that determines its topology, we introduce in this paper a notion ofG-uniformity for a groupoidG. The topology induced by aG-uniformity turnsGinto a topological locally transitive groupoid.

We also prove a Urysohn type lemma for groupoids and obtain metrization theorems for groupoids unifying in two ways the Alexandroff–Urysohn Theorem and Birkhoff- Kakutani Theorem.

1. Introduction and preliminaries

The notion of groupoid is a natural generalization of the notion of group in the following sense: a groupoid is a set G endowed with partially defined product operation (x, y)7→

xy

:G⁽²⁾ →G

(where G⁽²⁾ ⊂ G×G) and an inversion operation x 7→ x⁻¹ [:G→G]

satisfying the subsequent weaker versions of the group axioms:

G1 If (x, y) ∈ G⁽²⁾ and (y, z) ∈ G⁽²⁾, then (xy, z) ∈ G⁽²⁾, (x, yz) ∈ G⁽²⁾ and (xy)z = x(yz).

G2 (x⁻¹)⁻¹ =x for all x∈G.

G3 For all x∈G, (x, x⁻¹)∈G⁽²⁾, and if (z, x)∈G⁽²⁾, then (zx)x⁻¹ =z.

G4 For all x∈G, (x⁻¹, x)∈G⁽²⁾, and if (x, y)∈G⁽²⁾, thenx⁻¹(xy) = y.

The maps r and d on G, defined by the formulae r(x) = xx⁻¹ and d(x) = x⁻¹x, are called the range (target) map, respectively the domain (source) map. They have a common image called the unit space of G and denotedG⁽⁰⁾. The fibres of the range and the domain maps are denoted G^u =r⁻¹({u}) and G_v =d⁻¹({v}), respectively. Also for u, v ∈G⁽⁰⁾,G^u_v =G^u∩G_v.

A topological groupoid is a groupoid G together with a topology on G such that the product operation (x, y) 7→ xy

:G⁽²⁾ →G

(where G⁽²⁾ ⊂ G×G is endowed with the topology induced by the product topology on G×G) and the inversion operation

Received by the editors 2016-11-03 and, in final form, 2017-07-31.

Transmitted by Ronald Brown. Published on 2017-08-02.

2010 Mathematics Subject Classification: 22A22; 54E15, 54E35.

Key words and phrases: groupoid, Urysohn-type lemma, metrization theorem.

c M˘ad˘alina Roxana Buneci, 2017. Permission to copy for private use granted.

970

x 7→ x⁻¹ [:G→G] are continuous functions. A family {W_j}_j∈J of neighborhoods of the unit space is said to be compatible with the topology of the r-fibres (respectively, d-fibres) if for every u ∈ G⁽⁰⁾ and every open neighborhood U of u, there is j ∈ J such that W_j∩G^u ⊂U ∩G^u and u is in the interior of W_j ∩G^u with respect to the topology onG^u coming fromG(respectively,Wj∩Gu ⊂U∩Gu anduis in the interior of Wj∩Gu

with respect to the topology onG_u coming from G).

Let us also recall that a uniform space is a setX endowed with a uniform structure. A fundamental system of symmetric entourages of a uniform structure on X is a nonempty familyW of subsets of the Cartesian productX×X that satisfies the following conditions:

U1 if W is in W, thenW contains the diagonal ∆ ={(x, x) :x∈X}.

U2 if W₁ and W₂ are inW, then there is W₃ ∈ W such that W₃ ⊂W₁∩W₂.

U3 if W₁ is in W, then there exists W₂ inW such that, whenever (x, y) and (y, z) are in W₂, then (x, z)∈W₁.

U4 if W ∈ W, then W =W⁻¹ ={(y, x) : (x, y)∈W}(W is a symmetric entourage).

The uniform space X becomes a topological space by defining a subset A ⊂X to be open if and only if for everyx∈A there is W_x ∈ W such that {y: (x, y)∈W_x} ⊂A.

The Cartesian product X×X can be viewed as a trivial groupoid G under the oper- ations: (x, y) (y, z) = (x, z) and (x, y)⁻¹. In the settings of groupoids condition U1 can be written as ”G⁽⁰⁾ ⊂ W ⊂G for all W ∈ W” and condition U3 as ”for every W₁ ∈ W there is W₂ ∈ W such that W₂W₂ ⊂W₁”.

In this paper we work with a collection of subsets of a groupoid G mimicking the properties of fundamental system of symmetric entourages of a uniform structure on X.

Such a collection will be called in this paperG-uniformity. We prove that aG-uniformity induces a topology on G that turns G into a topological locally transitive groupoid. Let us recall that a topological locally transitive groupoid is a topological groupoidGwith the property that for all u∈G⁽⁰⁾ the maps r_u are open, where r_u :G_u →G⁽⁰⁾, r_u(x) =r(x) for allx∈Gu andGu is endowed with the topology coming fromG(see [12]). If we begin with a topological groupoid (G, τ) and with aG-uniformity given by a fundamental system of neighborhoods of the unit space, then the topology induced by deG-uniformity is finer than τ and coincides with τ if and only if (G, τ) is locally transitive. The main result of this paper is a Urysohn type lemma for groupoids (Theorem 2.5). The existence of a function with properties 1−3 in Theorem 2.5 could also be obtained taking into account that aG-uniformity is a base for a uniform structure onG. However the topology defined by the G-uniformity do not necessarily coincides with the groupoid topology, even if the G-uniformity is given by a fundamental system of neighborhoods of the unit space. The construction in Theorem 2.5 allows us to get a function with additional properties. In particular, in the case of a topological groupoid with open range map and aG-uniformity given by a fundamental system of neighborhoods of the unit space, our construction allows us to put out a connection with the groupoid topology: the functions f associated

in Theorem2.5 with open subsets of Gor with G⁽⁰⁾ are upper semi-continuous on Gand their restrictions to the r-fibres as well as to the d-fibres of the groupoid are continuous functions. Thus these functions can be used to construct convolutions algebras as in [4] and possibly to extend the construction of a C^∗-algebra associated to a topological locally compact groupoid with continuous Haar system introduced in [11]. Moreover the property 9 in Theorem 2.5 allows us to obtain metrization theorems for groupoids and thus to express in an unified way Alexandroff–Urysohn Theorem and Birkhoff-Kakutani Theorem as we explain below. Let us consider the following two theorems:

1.1. Theorem.[Alexandroff–Urysohn Theorem] A topological Hausdorff spaceX is me- trizable if and only if its topology is given by a uniformity with countable base. [1]

1.2. Theorem.[Birkhoff-Kakutani Theorem] A topological group Gis metrizable if and only if there is a countable base for the topology at identity element in G. Furthermore, in such a case, the distance function may be taken to be either left-invariant or right- invariant. ([2], [6])

Let us remark that the space X, respectively the group G can be viewed as r-fibres (as well asd-fibres) of a groupoid (X×X in the first case andGitself in the second case).

We prove in this paper that the previous two results can be express in an unified way in the groupoid language:

1.3. Theorem.Let G be a topological groupoid. Then there are left (respectively, right) invariant metrics compatible with the topology onr-fibres (respectively, thed-fibres) of the groupoid if and only if there is a countable G-uniformity {W_n}_n∈

N compatible with the topology of the r-fibres (respectively, d-fibres) such that T

n∈N

W_n=G⁽⁰⁾. (Proposition 3.14 and Proposition 3.15)

The proof of this theorem is based on the construction of a function on G satisfying the hypothesis of [8, Theorem 3.26]. This function is obtained as a particular case of Urysohn Lemma for groupoids (Theorem2.5).

We also prove in this paper that:

1.4. Theorem. For a topological locally transitive groupoid G the following statements are equivalent:

(a) G is metrizable

(b) For every neighborhoodW ofG⁽⁰⁾ there is a neighborhoodW⁰ ofG⁽⁰⁾ such thatW⁰W⁰ ⊂ W andG⁽⁰⁾ has a countable fundamental system{W_n}_n∈

Nof neighborhoods such that T

n∈N

W_n =G⁽⁰⁾ and T

n∈N

(r, d) (W_n) =diag G⁽⁰⁾ . (c) There is a countable G-uniformity{W_n}_n∈

N compatible with the topology of the fibres such that T

n∈N

W_n =G⁽⁰⁾ and T

n∈N

(r, d) (W_n) =diag G⁽⁰⁾

. Each W_n may be taken to be a neighborhood of the unit space.

Moreover the distance function ρ may be taken to satisfy the following properties:

1. ρ(x, y) =ρ(x⁻¹, y⁻¹) for all x, y ∈G.

2. ρ(x, r(x)) =ρ(x, d(x)) for all x∈G.

3. ρ(xy, r(x))≤ρ(x, r(x)) +ρ(y, r(y)) for all (x, y)∈G⁽²⁾. 4. ρ(x, y)≤ρ(x⁻¹y, d(x)) for all x, y ∈G such that r(x) = r(y).

5. ρ(d(x), d(y))≤2ρ(x, y) and ρ(r(x), r(y))≤2ρ(x, y) for all x,y ∈ G. (Theorem 3.16)

2. Urysohn’s lemma for groupoids

2.1. Definition.LetGbe a groupoid. By aG-uniformity we mean a collection{W}_W∈W of subsets of G satisfying the following conditions:

1. G⁽⁰⁾ ⊂W ⊂G for all W ∈ W.

2. If W1, W2 ∈ W, then there is W3 ⊂W1∩W2 such that W3 ∈ W.

3. For every W₁ ∈ W there is W₂ ∈ W such that W₂W₂ ⊂W₁. 4. W =W⁻¹for all W ∈ W.

2.2. Definition. Let G be a groupoid. Two G-uniformities W and W⁰ are said to be equivalent if for every W ∈ W there is W⁰ ∈ W⁰ such that W⁰ ⊂W and conversely, for every W⁰ ∈ W⁰ there is W ∈ W such that W ⊂W⁰.

LetWbe a family of subsets of a groupoidGsatisfying conditions 1−4 from Definition 2.1 and let

I = 1

2ⁿ, n ∈N

Let W₀ ∈ W and W₁ ∈ W be such that W₁W₁ ⊂ W₀. Inductively we can construct an I-indexed family {W_i}_i∈I. Suppose that for W_i ∈ W has already been built. Then according condition 3 in Definition 2.1, there is a W_i⁰ ∈ W such that W_i⁰W_i⁰ ⊂ W_i. Let W_i/2 = W_i⁰. Thus we obtain an I-indexed family {W_i}_i∈I satisfying the following properties:

1. Wi ∈ W for all i∈I.

2. W_iW_i ⊂W_2i for all i∈I, i≤ ¹₂. 3. W₁W₁ ⊂W₀.

Hence W_i ⊂W_iW_i ⊂W_2i for all i∈I, i≤ ¹₂ and

...W_1/2ⁿ ⊂W_1/2ⁿW_1/2ⁿ ⊂W_1/2ⁿ⁻¹ ⊂W_1/2ⁿ⁻¹W_1/2ⁿ⁻¹ ⊂...W_1/2 ⊂W_1/2W_1/2 ⊂W₁ Let us note that:

1. If i, j ∈I, theni < j iff there is p∈N^∗ such that j = 2^pi.

2. If i, j ∈I and i < j, then 2i≤j.

3. If i, j ∈I and i≤j, then W_i ⊂W_j.

4. If i₁, i₂, ..., i_k ∈ I and i_k ≤i_k−1 < i_k−2 < ... < i₁ <1, then W_ikW_ik−1...W_i1 ⊂W_2i1 and W_i1...W_ik−1W_ik ⊂W_2i1. Indeed,

W_ikW_ik−1W_ik−2...W_i1 ⊂ W_ik−1W_ik−1W_ik−2...W_i1

⊂ W_2ik−1W_ik−2...W_i1

⊂ W_ik−2W_ik−2...W_i1

⊂ ...⊂W_2i1

Similarly, W_i1...W_ik−1W_ik ⊂W_i1...W_ik−1W_ik−1 ⊂W_i1...W_ik−2W_2ik−1 ⊂...W_2i1.

5. If i₁, i₂, ..., i_k, j₁, j₂, ..., j_m ∈ I, i_k < ik−1 < ... < i₁ ≤ 1,j_m < jm−1 < ... < j₁ ≤ 1 and i_k+ik−1+...+i₁ ≤j_m+jm−1+...+j₁, then

W_ikW_ik−1...W_i1 ⊂W_jmW_jm−1...W_j1.

Indeed, let us remark thati_k+ik−1+...+i₁ = _2nk¹ +_2nk−1¹ +...+₂n¹1 is the conversion into decimal system of the following number in base 2: b₀, b₁b₂...b_nk where b_i = 1 if i∈ {n1, n2, ..., nk} andbi = 0 otherwise. Thus ifik+ik−1+...+i1 =jm+jm−1+...+j1, thenm =k and i_k =j_k, ..., i₁ =j₁. Ifi_k+ik−1+...+i₁ < j_m+jm−1+...+j₁, then there isp∈N^∗ such that i₁ =j₁, ..., ip−1 =jp−1 and i_p < j_p. Hence

W_ikW_ik−1...W_ipW_ip−1... W_i1 ⊂ W_2ipW_ip−1... W_i1

⊂ W_jpW_ip−1... W_i1

= W_jpW_jp−1... W_j1

⊂ W_jmW_jm−1...W_j1.

2.3. Lemma.Let G be a groupoid, W be a G-uniformity (in the sense of Definition 2.1) and let

I = 1

2ⁿ, n∈N

.

Let us consider an I-indexed family {W_i}_i∈I satisfying the following properties:

1. Wi ∈ W for all i∈I.

2. W_iW_i ⊂W_2i for all i∈I, i≤ ¹₂.

For W_ik, W_ik−1, ..., W_i1 ∈ {W_i}_i∈I, let us denote s W_ikW_ik−1...W_i1

=i_k+ik−1+...+i₁.

Let n ∈N^∗, and i_1, i₂, ..., i_k ∈I be such that i_k < ik−1 < ik−2 < ... < i₁ <1. Then there are j₁, j₂, ..., j_r ∈I such that

1. j_r < j_r−1 < i_r−2 < ... < j₁ ≤max ₁

2ⁿ⁻¹,2i₁ ≤1 2. W_1/2ⁿW_ikW_ik−1...W_i1 ⊂W_jrW_jr−1...W_j

1

3. 0< s WjrWjr−1...Wj₁

−s Wi_kWik−1...Wi1

≤ ₂n−1¹

Moreover j₁ < s W_ikW_ik−1...W_i1

+ ₂n−1¹ and if j_r 6= ₂¹n, then j_r ≥ ₂n−1¹ . Also if

1

2ⁿ ≤i_k, then s W_jrW_jr−1...W_j

1

−s W_ikW_ik−1...W_i1

≤ ₂¹n.

Proof. Case 1: ₂¹n < i_k. Obviously, ₂¹n < i_k < ik−1 < ik−2 < ... < i₁ < 1 and s W_1/2ⁿW_ikW_ik−1...W_i1

=s W_ikW_ik−1...W_i1 + ₂¹n.

Case 2: There is m ∈ {2,3, ..., k} such thati_m = ₂¹n < ^im−1₂ . Then W_1/2ⁿW_ikW_ik−1...W_imW_im−1...W_i1 ⊂W_1/2ⁿW_2imW_im−1...W_i1. and we have

s W_1/2ⁿW_2imW_im−1...W_i1

=

= s W_ik...W_imW_im−1...W_i1

−(i_k+...+i_m) + 2i_m+ 1 2ⁿ

≤ s W_ik...W_imW_im−1...W_i1

+i_m+ 1 2ⁿ

= s W_ik...W_imW_im−1...W_i1 + 2

2ⁿ. Moreover i_k +...+i_m ≤ ₂k−m¹ +₂k−m¹ +...¹₂ + 1

i_m < 2i_m < 2i_m + ₂¹n. Consequently, s W_1/2ⁿW_2imW_im−1...W_i1

> s W_ik...W_imW_im−1...W_i1 .

Case 3: There is m ∈ {2,3, ..., k} such that i_m = ₂¹n = ^im−1₂ and there is q ∈ {2,3, ..., m−1} such that 4i_q ≤iq−1. Let p be the greatest element of the set

{q: 2 ≤q≤m−1, 4i_q ≤i_q−1}.

Then W_1/2ⁿW_ik...W_i1 ⊂ W_1/2ⁿW_2imW_im−1...W_i1 ⊂ W_1/2ⁿW_2ipW_ip−1...W_i1. Moreover s W2ipWip−1...Wi1

=s Wim−1...WipWip−1...Wi1

−(im−1 +..+ip) + 2ip

= s W_im−1...W_ipW_ip−1...W_i1

−(im−1+ 2im−1+...+ 2^m−p−1im−1) + 2^m−pim−1

= s W_im−1...W_ipW_ip−1...W_i1

−im−1 2^m−p−1

+im−12^m−p

= s W_ik...W_ipW_ip−1...W_i1

−(i_k+...+i_m) +im−1, and since ₂n−1¹ =im−1, it follows that

s W_1/2ⁿW_2ipW_ip−1...W_i1

=s W_2ipW_ip−1...W_i1 + 1

2ⁿ

= s W_ikW_ik−1...W_i1

−(i_k+...+i_m) +im−1+ 1 2ⁿ

= s W_ikW_ik−1...W_i1

−(i_k+...+i_m+1)− 1

2ⁿ + 1 2ⁿ⁻¹ + 1

2ⁿ

= s W_ikW_ik−1...W_i1

−(i_k+...+i_m+1) + 1 2ⁿ⁻¹. On the other hand, i_k +... +i_m ≤ ₂k−m+1¹ + ₂k−m¹ +...¹₂

im−1 < im−1 and therefore s W_1/2ⁿW_2ipW_ip−1...W_i1

> s W_ikW_ik−1...W_i1 .

Case 4: There is m∈ {2,3, ..., k}such that i_m = ₂¹n = ^im−1₂ = ^im−2₂2 =...= ₂m−1ⁱ¹ . Then W_1/2ⁿW_ik...W_i1 ⊂ W_1/2ⁿW_2imW_im−1...W_i1 ⊂ W_1/2ⁿW_2i1 and

s W_1/2ⁿW_2i1

= s(W_im...W_i1)−(i_m+..+i₁) + 2i₁+ 1 2ⁿ

= s(W_im...W_i1)−(i_m+ 2i_m+...+ 2^m−1i_m) + 2^mi_m+ 1 2ⁿ

= s(W_im...W_i1)−i_m(2^m−1) +i_m2^m+ 1 2ⁿ

= s W_ik...W_im+1W_im...W_i1

−(i_k+...+im−1) +i_m+ 1 2ⁿ

< s W_ikW_ik−1...W_i1

+ 1

2ⁿ⁻¹. Also s W_1/2ⁿW_2i1

= ₂¹n + 2i₁ >2i₁ > s W_ikW_ik−1...W_i1 and j₁ = 2i₁ < s W_ikW_ik−1...W_i1

+ 1

2ⁿ⁻¹. Case 5: There is m ∈ {2,3, ..., k} such thatim < ₂¹n < ^im−1₂ . Then

W1/2ⁿWi_kWik−1...WimWim−1...Wi1 ⊂ W1/2ⁿW2imWim−1...Wi1

⊂ W_1/2ⁿW_1/2ⁿW_im−1...W_i1

⊂ W_1/2ⁿ⁻¹W_im−1...W_i1. and

s W_1/2ⁿ⁻¹W_im−1...W_i1

= s W_ik...W_imW_im−1...W_i1

−(i_k+...+i_m) + 1 2ⁿ⁻¹

< s W_ikW_ik−1...W_i1

+ 1

2ⁿ⁻¹.

Moreover i_k+...+i_m ≤ ₂k−m¹ + ₂k−m¹ +...¹₂ + 1

i_m < 2i_m ≤ ₂¹n < ₂n−1¹ . Consequently, s W_1/2ⁿ⁻¹Wim−1...Wi1

> s Wi_k...WimWim−1...Wi1

.

Case 6: There is m ∈ {2,3, ..., k} such that i_m < ₂¹n = ^im−1₂ and there is q ∈ {2,3, ..., m−1} such that 4i_q ≤i_q−1. If p is the greatest element of the set

{q: 2 ≤q≤m−1, 4i_q ≤iq−1}, then

W_1/2ⁿW_ikW_ik−1...W_imW_im−1...W_i1 ⊂ W_1/2ⁿW_2imW_im−1...W_i1

⊂ W_1/2ⁿW_1/2ⁿW_im−1...W_i1

⊂ W_1/2ⁿ⁻¹W_im−1...W_i1

⊂ W2ipWip−1...Wi1. Moreover

s W_2ipW_ip−1...W_i1

=s W_im−1...W_ipW_ip−1...W_i1

−(im−1 +..+i_p) + 2i_p

= s W_im−1...W_ipW_ip−1...W_i1

−(i_m−1+ 2i_m−1+...+ 2^m−p−1i_m−1) + 2^m−pi_m−1

= s Wim−1...WipWip−1...Wi1

−im−1 2^m−p−1

+im−12^m−p

= s Wi_k...WimWim−1...Wi1

−(ik+...+im) +im−1

= s Wi_kWik−1...Wi1

−(ik+...+im) + 1 2ⁿ⁻¹. Hence

s W_2ipW_ip−1...W_i1

< s W_ikW_ik−1...W_i1

+ 1

2ⁿ⁻¹. Since we have ik+...+im ≤ ₂k−m+1¹ +₂k−m¹ +...¹₂

im−1 < im−1, it follows that im−1− (i_k+...+i_m) > 0. Thus s W_2ipW_ip−1...W_i1

> s W_ik...W_imW_im−1...W_i1

. We also have y_r= 2i_p ≥ ₂n−1¹ .

Case 7: There is m∈ {2,3, ..., k}such that i_m < ₂¹n = ^im−1₂ = ^im−2₂2 =...= ₂m−1ⁱ¹ . Then W_1/2ⁿWi_k...Wi1 ⊂W_1/2ⁿW2imWim−1...Wi1 ⊂W_1/2ⁿ⁻¹Wim−1...Wi1W_1/2ⁿW2i1 ⊂...W2i1 and

s(W_2i1) = s W_im−1...W_i1

−(im−1+..+i₁) + 2i₁

= s W_im−1...W_i1

−(im−1+ 2im−1+...+ 2^m−2im−1) + 2^m−1im−1

= s Wim−1...Wi1

−im−1 2^m−1−1

+im−12^m−1+ 1 2ⁿ

= s W_ik...W_imW_im−1...W_i1

−(i_k+...+i_m) +im−1

< s W_ikW_ik−1...W_i1

+ 1

2ⁿ⁻¹.

Also s(W_2i1) = 2i₁ > ₂k−1¹ +...+ ¹₂ + 1

i₁ ≥ i_k +ik−1 +...i₁ ≥ s W_ikW_ik−1...W_i1 . Moreover j1 = 2i1 =s(W2i1)< s Wi_kWik−1...Wi1

+₂n−1¹ . Case 8: ₂¹n =i₁. We have

W_1/2ⁿW_ikW_ik−1...W_i1 ⊂W_1/2ⁿW_2i1,

s W_1/2ⁿW_2i1

= 1

2ⁿ + 2i₁ ≤s W_ikW_ik−1...W_i1 + 1

2ⁿ +i₁

= s W_ikW_ik−1...W_i1

+ 1

2ⁿ⁻¹. and ₂¹n + 2i₁ > 2i₁ > ₂k−1¹ +...+¹₂ + 1

i₁ ≥ i_k+ik−1+...i₁ = s W_ikW_ik−1...W_i1 . We also have j1 = 2i1 < s W1/2ⁿW2i1

≤s Wi_kWik−1...Wi1

+ ₂n−1¹ . Case 9: ₂¹n > i₁. We have

W_1/2ⁿW_ikW_ik−1...W_i1 ⊂W_1/2ⁿW_2i1 ⊂W_1/2ⁿW_1/2ⁿ ⊂W_1/2ⁿ⁻¹, s W_1/2ⁿ⁻¹

= 1

2ⁿ⁻¹ < s W_ikW_ik−1...W_i1

+ 1

2ⁿ⁻¹ and ₂n−1¹ >2i₁ > ₂k−1¹ +...+¹₂ + 1

i₁ ≥i_k+ik−1+...i₁ =s W_ikW_ik−1...W_i1

. Moreover j₁ = ₂n−1¹ < s W_ikW_ik−1...W_i1

+₂n−1¹ .

Let us also remark that if ₂¹n = i_k , then W_1/2ⁿW_ikW_ik−1...W_i1 ⊂ W_2imW_im−1...W_i1, where m is the greatest element of the set {q: 2 ≤q≤k, 4i_q ≤iq−1} if the set is not empty or m= 1, otherwise. We have

s W_2imW_im−1...W_i1

= s W_ik...W_imW_im−1...W_i1

−(i_k+...+i_m) + 2i_m

= s W_ikW_ik−1...W_i1

− 1 + 2 +...+ 2^k−m 1

2ⁿ +2^k−m+1 2ⁿ

= s W_ikW_ik−1...W_i1 + 1

2ⁿ. Moreover i_k+...+i_m ≤ ₂k−m¹ +₂k−m¹ +...¹₂ + 1

i_m <2i_m. Consequently, s W_2imW_im−1...W_i1

> s W_ik...W_imW_im−1...W_i1 .

2.4. Remark. In the preceding lemma since W_1/2ⁿWi_kWik−1...Wi1 ⊂ WjrWjr−1...Wj₁, it follows that W_1/2ⁿW_ikW_ik−1...W_i1−1

⊂ W_jrW_jr−1...W_j

1

−1

and consequently, W_i1W_i2...W_ikW_1/2ⁿ ⊂W_j1W_j2...W_jr.

2.5. Theorem.LetGbe a groupoid,W be aG-uniformity (in the sense of Definition2.1) and letW ∈ W. Let us consider anI = ₁

2ⁿ, n∈N -indexed subfamilyW_I ={W_i}_i∈I of W as in Lemma2.3 such that W₁ ⊂W. Then for every subsetA of Gthere is a function f =fA,W_I :G→[0, 1] satisfying the following conditions:

1. If n ∈N, n ≥2, x∈G and y∈W_1/2ⁿxW_1/2ⁿ, then |f(x)−f(y)|< ₂n−2¹ . 2. f(x) = 0 for all x∈A.

3. f(x) = 1 for all x /∈W AW.

4. If A=A⁻¹, then f(x) = f(x⁻¹) for all x∈G.

5. If G is endowed with a topology such that Wi_kWik−1...Wi1A Wi1...Wik−1Wi_k is open for all i₁, i₂, ..., i_k ∈I, i_k < ik−1 < ... < i₁ <1, then f is upper semi-continuous.

6. For all n∈N, n≥2, we have W_1/2ⁿ⁺¹AW_1/2ⁿ⁺¹ ⊂

x:f(x)< 1 2ⁿ

⊂W_1/2ⁿ⁻¹AW_1/2ⁿ⁻¹.

In particular, if A =G⁽⁰⁾, then W_1/2ⁿ⁺¹W_1/2ⁿ⁺¹ ⊂

x:f(x)< 1 2ⁿ

⊂W_1/2ⁿ⁻¹W_1/2ⁿ⁻¹ ⊂W_1/2ⁿ⁻²

for all n∈N, n≥2.

7. If A=G⁽⁰⁾, then f(xy)≤3f(x) +f(y) for all (x, y)∈G⁽²⁾. 8. If A=G⁽⁰⁾, then f(xy)≤2 (f(x) +f(y)) for all (x, y)∈G⁽²⁾.

9. If A=G⁽⁰⁾, then f(x₁x₂...x_n)≤3 (f(x₁) +f(x₂) +...+f(x_n)) for all n ∈N and x₁, x₂, ..., x_n ∈G such that d(x_i) =r(x_i+1) for all i∈ {1,2, ..., n−1}.

10. If A = G⁽⁰⁾ and for every x ∈ G\ G⁽⁰⁾ there is i_x ∈ I such that x /∈ W_ix (or equivalently, T

n

W_1/2ⁿ =G⁽⁰⁾), then f⁻¹({0}) = G⁽⁰⁾. Proof.For each x∈G, let us define

i(x) = inf

i_k+ik−1+...+i₁ : i₁, i₂, ..., i_k ∈I, i_k< ik−1 < ... < i₁, x∈Wi_kWik−1...Wi1A Wi1...Wik−1Wi_k

(with convention inf∅=∞) and

f(x) = min{i(x),1}.

1. Letx ∈Gand y∈W_1/2ⁿxW_1/2ⁿ. If i(x)≥ 1 and i(y)≥1, thenf(x) =f(y) = 1.

Let us suppose that i(x)<1 or i(y)<1.

Case 1: i(x) < 1. Then there are i₁, i₂, ..., i_k ∈ I, i_k < i_k−1 < ... < i₁ < 1 such that x ∈ W_ikW_ik−1...W_i1A W_i1...W_ik−1W_ik and i_k+ik−1 +...+i₁ < i(x) + ₂¹n. By Lemma 2.3, there are j₁, j₂, ..., j_r ∈ I, j_r < jr−1 < ir−2 < ... < j₁ ≤ 1 such that W_1/2ⁿW_ikW_ik−1...W_i1 ⊂W_jrW_jr−1...W_j

1 and

0<(j_r+...+j₁)−(i_k+i_k−1+...+i₁)< 3 2ⁿ Hence

ik+ik−1+...+i1 ≤jr+...+j1 < i(x) + 1 2ⁿ⁻² and since

y ∈ W_1/2ⁿxW_1/2ⁿ ⊂

⊂ W_1/2ⁿW_ikW_ik−1...W_i1AW_i1...W_ik−1W_ikW_1/2ⁿ

⊂ W_jrW_jr−1...W_j

1AW_j1W_j2...W_jr

it follows that i(y)< i(x) +₂n−2¹ . If i(y)<1, then sincey∈W_1/2ⁿxW_1/2ⁿ is equivalently to x ∈ W_1/2ⁿyW_1/2ⁿ it follows that i(x) < i(y) + ₂n−2¹ . Therefore |f(x)−f(y)| =

|i(x)−i(y)| < ₂n−2¹ . If i(y) ≥ 1, then |f(x)−f(y)| =|i(x)−1| = 1−i(x) ≤ i(y)− i(x)< ₂n−2¹ .

Case 2: i(y)<1. Since y∈W_1/2ⁿxW_1/2ⁿ is equivalently tox∈W_1/2ⁿyW_1/2ⁿ, the case i(y)<1 can be treated similarly as the case i(x)<1.

2. Let us prove that f(x) = 0 for all x ∈ A. Since A ⊂ W_1/2ⁿAW_1/2ⁿ for all n, it follows that i(x) = 0, and consequently, f(x) = 0 for all x∈A.

3.Let us prove thatf(x) = 1 for allx /∈W AW. Letx /∈W AW. By contradiction, let us supposef(x)<1. We necessarily havei(x)<1,and hence there arei₁, i₂, ..., i_k∈I, i_k < ik−1 < ... < i₁ <1 such that

x ∈ W_ikW_ik−1...W_i1A W_i1...W_ik−1W_ik ⊂W_2i1AW_2i1

⊂ W₁AW₁ ⊂W AW

This is in contradiction to the hypothesis x /∈W AW. 4. Since A=A⁻¹, it follows that

W_ikW_ik−1...W_i1A W_i1...W_ik−1W_ik−1

=W_ikW_ik−1...W_i1A W_i1...W_ik−1W_ik. Thusx∈W_ikW_ik−1...W_i1A W_i1...W_ik−1W_ik if and only if

x⁻¹ ∈W_ikW_ik−1...W_i1A W_i1...W_ik−1W_ik. Therefore f(x) =f(x⁻¹) for all x∈G.

5. Letα ∈Rand let us consider the set

U_α ={x∈G:f(x)< α}.

If α > 1, then U_α = G, hence U_α is an open set. Let us consider α ≤ 1 and let x ∈ U_α. Then f(x) < 1. Thus i(x) < 1, and hence there are i₁, i₂, ..., i_k ∈ I, i_k < ik−1 < ... < i₁ <1 such that

x∈Wi_kWik−1...Wi1A Wi1...Wik−1Wi_k

i_k+ik−1+...+i₁ < α.

For all y∈W_ikW_ik−1...W_i1A W_i1...W_ik−1W_ik we have i(y)< α. Consequently, x∈W_ikW_ik−1...W_i1A W_i1...W_ik−1W_ik ⊂U_α.

Therefore U_α is open.

6. Ifx∈W_1/2ⁿ⁺¹AW_1/2ⁿ⁺¹, then i(x)≤ ₂n+1¹ . Thus f(x)≤ ₂n+1¹ < ₂¹n. If f(x)< ₂¹n <

1, then i(x) < ₂¹n and there are i₁, i₂, ..., i_k ∈ I, i_k < ik−1 < ... < i₁ < 1 such that x ∈ W_ikW_ik−1...W_i1A W_i1...W_ik−1W_ik and i_k+ik−1 +...+i₁ < i(x) + ₂¹n < ₂n−1¹ . Hence i₁ < ₂n−1¹ and therefore x∈W_2i1AW_2i1 ⊂W_1/2ⁿ⁻¹AW_1/2ⁿ⁻¹.

7. Let (x, y) ∈ G⁽²⁾. If 3f(x) + f(y) ≥ 1, then obviously, f(xy) ≤ 3f(x) +f(y).

Let us suppose that 3f(x) +f(y) < 1 or equivalently, 3i(x) +i(y) < 1 (consequently, i(x) < ¹₃ and i(y) < 1). Let ε > 0 such that ε < 1−3i(x)−i(y). Then there are i₁, i₂, ..., i_k ∈ I, i_k < ik−1 < ... < i₁ ≤ ¹₄ such that x ∈ W_ikW_ik−1...W_i1W_i1...W_ik−1W_ik, i_k+ik−1 +...+i₁ < i(x) + ^ε₃ and there are j₁, j₂, ..., j_m ∈ I, j_m < jm−1 < ... < j₁ ≤ ¹₂ such that y ∈ W_jmW_jm−1...W_j1W_j1...W_jm−1W_jm, j_m +j_m−1 + ...+j₁ < i(y) + ^ε₃. By Lemma 2.3, there are q₁¹, q₂¹, ..., q_r¹₁ ∈ I, q_r¹₁ < q_r¹₁₋₁ < q¹_r1−2 < ... < q₁¹ ≤ 1 such that W_ikW_jmW_im−1...W_j1 ⊂W_qr¹

1W_q¹

r1−1...W_q¹

1, 0< q¹_r

1 +...+q₁¹

−(j_m+j_m−1+...+j₁)≤2i_k.

andq¹₁ ≤j_m+jm−1+...+j₁+2i_k < i(y)+^ε₃+2i(x)+^2ε₃ <1. Repeatedly applying Lemma 2.3, for p= 2,3, ..., k there areq₁^p, q^p₂, ..., q_rp^p ∈I, q_r^p

p < q^p_r

p−1 < q_r^p

p−2 < ... < q₁^p ≤ 1 such that W_ik−p+1W_q^p=1

rp=1W_q^p−1

rp−1−1...W_q^p−1

1 ⊂W_q^p_rpW_q^p

rp−1...W_q^p

1 , 0≤

q_r^p_p+...+q₁^p

−

q_r^p−1_p−1 +...+q^p−1₁

≤2ik−p+1. and

q₁^p−1 < q^p−1_rp−1 +...+q₁^p−1+ 2ik−p+1

< q^p−2_rp−2 +...+q₁^p−2+ 2ik−p+ 2ik−p+1

...

< jm+jm−1+...+j1 + 2ik+...+ 2ik−p+ 2ik−p+1

< i(y) + ε

3 + 2i(x) + 2ε 3 <1.