����������� ������ 50 (1998), 17–22

UDK 515.124.2

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THE COMPATIBILITY OF THE TANGENCY RELATIONS OF SETS IN GENERALIZED METRIC SPACES

Tadeusz Konik

Abstract. In this paper the problem of the compatibility of the tangency relations T_{li}(a_{i}, b_{i}, k, p) (i = 1, 2) of sets of the classes M̃_{p,k} and A_{p,k}^{∗} in a generalized metric space is considered. Some sufficient conditions for the compatibility of these relations of sets of the above classes are given here.

Introduction

Let E be an arbitrary non-empty set and let l be a non-negative real function defined on the Cartesian product E_{0} ×E_{0} of the family E_{0} of all non-empty subsets of the set E. Let l_{0} be the function defined by the formula

l_{0}(x, y) = l({x}, {y}) for x, y ∈ E. (1)

If we put some conditions on the function l, then the function l_{0} defined by (1) will be a metric on the set E. By this reason the pair (E, l) can be treated as a certain generalization of a metric space and we shall call it (see [9]) a generalized metric space. Using (1) we may define in the space (E, l), similarly as in a metric space, the following notions: the sphere S_{l}(p, r) and the ball K_{l}(p, r) with the centre at the point p and the radius r.

Let S_{l}(p, r)_{u} denote the so-called u-neighbourhood of the sphere S_{l}(p, r) in the space (E, l) defined by the formula

S_{l}(p, r)_{u} = \{ \begin{matrix}⋃_{q∈S_{l}(p,r)} K_{l}(q, u), & for u > 0, \\ S_{l}(p, r), & for u = 0. \end{matrix}  (2)

Let a, b be arbitrary non-negative real functions defined in a certain right-hand side neighbourhood of 0 such that

a(r) ⟶_{r→0+} 0 and b(r) ⟶_{r→0+} 0.  (3)

We say that the pair (A, B) of sets A, B of the family E_{0} is (a, b)-clustered at the point p of the space (E, l), if 0 is the cluster point of the set of all real numbers r > 0 such that the sets of the form A ∩ S_{l}(p, r)_{a(r)} and B ∩ S_{l}(p, r)_{b(r)} are non-empty.

AMS Subject Classification: 53 A 99

Keywords and phrases: generalized metric space, compatibility of the tangency of sets

Let (see [9])

T_{l}(a, b, k, p) = { (A, B) : A, B ∈ E_{0}, the pair (A, B) is (a, b)-clustered at the

point p of the space (E, l) and \frac{1}{r^{k}} l(A ∩ S_{l}(p, r)_{a(r)}, B ∩ S_{l}(p, r)_{b(r)}) ⟶_{r→0+} 0 \}. (4)

If (A, B) ∈ T_{l}(a, b, k, p) then we say that the set A ∈ E_{0} is (a, b)-tangent of order k (k > 0) to the set B ∈ E_{0} at the point p of the space (E, l).

We shall call T_{l}(a, b, k, p) defined by (4) the relation of (a, b)-tangency of order k at the point p, or shortly: the tangency relation of sets in the generalized metric space (E, l).

Two relations of the tangency T_{l_{1}}(a_{1}, b_{1}, k, p) and T_{l_{2}}(a_{2}, b_{2}, k, p) are called compatible if (A, B) ∈ T_{l_{1}}(a_{1}, b_{1}, k, p) if and only if (A, B) ∈ T_{l_{2}}(a_{2}, b_{2}, k, p) for A, B ∈ E_{0}.

We say that the set A ∈ E_{0} has the Darboux property at the point p of the space (E, l), which we write: A ∈ D_{p}(E, l) (see [3]), if there exists a number τ > 0 such that the set A ∩ S_{l}(p, r) is non-empty for r ∈ (0, τ).

Let ρ be a metric on the set E and let A, B be arbitrary sets of the family E_{0}. Let us denote

ρ(A, B) = inf{ ρ(x, y) : x ∈ A, y ∈ B }, d_{ρ}A = sup{ ρ(x, y) : x, y ∈ A }. (5)

Let f be a subadditive increasing real function defined in a certain right-hand side neighbourhood of 0 such that f(0) = 0. By \overline{F}_{f} we shall denote the class of all functions l fulfilling the conditions:

1^{∘} l: E_{0} × E_{0} → ⟨0, ∞),

2^{∘} f(ρ(A, B)) ⩽ l(A, B) ⩽ f(d_{ρ}(A ∪ B)) for A, B ∈ E_{0}.

Since

f(ρ(x, y)) = f(ρ({x}, {y})) ⩽ l({x}, {y}) ⩽ f(d_{ρ}({x} ∪ {y})) = f(ρ(x, y)),

then from here and from (1) it follows that

l_{0}(x, y) = l(\{x\}, \{y\}) = f(ρ(x, y)) for l ∈ \overline{F}_{f} and x, y ∈ E.  (6)

It is easy to prove that the function l_{0} defined by (6) is a metric on the set E.

In the present paper the problem of the compatibility of the tangency relations of sets of the classes M̃_{p,k} and A_{p,k}^{∗} having the Darboux property at the point p of the space (E, l), for the functions l belonging to the class \overline{F}_{f}, is considered.

1. On the compatibility of the tangency relations of sets of the classes M̃_{p,k}

By A′ we shall denote the set of all cluster points of the set A ∈ E_{0}. Let k be any fixed positive real number and let

ρ(x, A) = inf{ ρ(x, y) : y ∈ A }.  (1.1)

Let us put by definition (see [4])

M̃_{p,k} = { A ∈ E_{0} : p ∈ A′ and there exists μ > 0 scuh that for an arbitrary ε > 0

there exists δ > 0 such that for every pair of points (x, y) ∈ [A, p; μ, k]

if ρ(x, y) < δ and \frac{ρ(x, A)}{ρ^{k}(x, p)} < δ then \frac{ρ(x, y)}{ρ^{k}(x, p)} < ε \},  (1.2)

where

[A, p; μ, k] = { (x, y) : x ∈ E, y ∈ A and μρ(x, A) < ρ^{k}(x, p) = ρ^{k}(y, p) } (1.3)

In the paper [4] the following lemma was proved.

Lemma 1.1. If

\frac{a(r)}{r^{k+1}} ⟶_{r→0+} α,  (1.4)

where α < ∞, then for an arbitrary set A ∈ M̃_{p,k} ∩ D_{p}(E, ρ)

\frac{1}{r^{k}} d_{ρ}(A ∩ S_{ρ}(p, r)_{a(r)}) ⟶_{r→0+} 0.  (1.5)

From this lemma and from the fact that every function l ∈ \overline{F}_{f} generates on the set E the metric defined by the formula (6) it follows that

\frac{1}{r^{k}} d_{l}(A ∩ S_{l}(p, r)_{a(r)}) ⟶_{r→0+} 0,  (1.6)

for l ∈ \overline{F}_{f} and A ∈ M̃_{p,k} ∩ D_{p}(E, l), when the function a fulfills the condition (1.4).

Theorem 1.1. If l_{i} ∈ \overline{F}_{f} (i = 1, 2),
\frac{a(r)}{r^{k+1}} ⟶_{r→0+} α and \frac{b(r)}{r^{k+1}} ⟶_{r→0+} β,  (1.7)

where α, β < ∞, then for arbitrary sets of the classes M̃_{p,k} ∩ D_{p}(E, l) the tangency relations T_{l_{1}}(a, b, k, p) and T_{l_{2}}(a, b, k, p) are compatible.

Proof. Let us assume that (A, B) ∈ T_{l_{1}}(a, b, k, p) for A, B ∈ M̃_{p,k}. Hence, from (2) and from the fact that (see (6))

l_{1}({x}, {y}) = l_{2}({x}, {y}) = l_{0}(x, y) for x, y ∈ E,  (1.8)

it follows that the pair of sets (A, B) is (a, b)-clustered at the point p of the space (E, l_{1}) and

\frac{1}{r^{k}} l_{1}(A ∩ S_{l}(p, r)_{a(r)}, B ∩ S_{l}(p, r)_{b(r)}) ⟶_{r→0+} 0.  (1.9)

From the inequality

d_{ρ}(A ∪ B) ⩽ d_{ρ}A + d_{ρ}B + ρ(A, B) for A, B ∈ E_{0},  (1.10)

from the properties of the function f and from the fact that l_{1}, l_{2} ∈ \overline{F}_{f} we get

|\frac{1}{r^{k}} l_{2}(A ∩ S_{l}(p, r)_{a(r)}, B ∩ S_{l}(p, r)_{b(r)}) − \frac{1}{r^{k}} l_{1}(A ∩ S_{l}(p, r)_{a(r)}, B ∩ S_{l}(p, r)_{b(r)})| ⩽

\frac{1}{r^{k}} f(d_{ρ}((A∩S_{l}(p, r)_{a(r)})∪(B∩S_{l}(p, r)_{b(r)})))− \frac{1}{r^{k}} f(ρ(A∩S_{l}(p, r)_{a(r)}, B∩S_{l}(p, r)_{b(r)}))

⩽ \frac{1}{r^{k}} f(d_{ρ}(A∩S_{l}(p, r)_{a(r)})+d_{ρ}(B∩S_{l}(p, r)_{b(r)})+ρ(A∩S_{l}(p, r)_{a(r)}, B∩S_{l}(p, r)_{b(r)}))−

− \frac{1}{r^{k}} f(ρ(A ∩ S_{l}(p, r)_{a(r)}, B ∩ S_{l}(p, r)_{b(r)}))

⩽ \frac{1}{r^{k}} f(d_{ρ}(A ∩ S_{l}(p, r)_{a(r)})) + \frac{1}{r^{k}} f(d_{ρ}(B ∩ S_{l}(p, r)_{b(r)})). (1.11)

From the fact that f is an increasing function we obtain

f(d_{ρ}(A ∩ S_{l}(p, r)_{a(r)})) = f(sup{ ρ(x, y) : x, y ∈ (A ∩ S_{l}(p, r)_{a(r)}) })

= sup{ f(ρ(x, y)) : x, y ∈ (A ∩ S_{l}(p, r)_{a(r)}) } = sup{ l_{0}(x, y) : x, y ∈ (A ∩ S_{l}(p, r)_{a(r)}) } = d_{l}(A ∩ S_{l}(p, r)_{a(r)}). (1.12)

Hence and from (1.6) it follows that

\frac{1}{r^{k}} f(d_{ρ}(A ∩ S_{l}(p, r)_{a(r)})) ⟶_{r→0+} 0.  (1.13)

Analogously

\frac{1}{r^{k}} f(d_{ρ}(B ∩ S_{l}(p, r)_{b(r)})) ⟶_{r→0+} 0.  (1.14)

From (1.9), (1.13), (1.14) and from the inequality (1.11) we get

\frac{1}{r^{k}} l_{2}(A ∩ S_{l}(p, r)_{a(r)}, B ∩ S_{l}(p, r)_{b(r)}) ⟶_{r→0+} 0.  (1.15)

Since the functions l_{1}, l_{2} ∈ \overline{F}_{f} generate on the set E the same metric l_{0} (see (6)), from the fact that the pair of sets (A, B) is (a, b)-clustered at the point p of the space (E, l_{1}), it follows that it is (a, b)-clustered at the point p of the space (E, l_{2}). Hence and from (1.15) it results that (A, B) ∈ T_{l_{2}}(a, b, k, p).

If (A, B) ∈ T_{l_{2}}(a, b, k, p), then similarly we prove that (A, B) ∈ T_{l_{1}}(a, b, k, p). Hence it follows that the tangency relations T_{l_{1}}(a, b, k, p) and T_{l_{2}}(a, b, k, p) are compatible in the classes of sets M̃_{p,k} ∩ D_{p}(E, l).

Let a_{i}, b_{i} (i = 1, 2) be non-negative real functions defined in a certain righthand side neighbourhood of 0 and fulfilling the condition

a_{i}(r) ⟶_{r→0+} 0 and b_{i}(r) ⟶_{r→0+} 0.  (1.16)

In the paper [7] the following theorem was proved.

Theorem 1.2. If l ∈ \overline{F}_{f} and
\frac{a_{i}(r)}{r^{k+1}} ⟶_{r→0+} α_{i}, \frac{b_{i}(r)}{r^{k+1}} ⟶_{r→0+} β_{i},  (1.17)

where α_{i}, β_{i} < ∞ for i = 1, 2, then for arbitrary sets of the classes M̃_{p,k} ∩ D_{p}(E, l) the tangency relations T_{l}(a_{1}, b_{1}, k, p) and T_{l}(a_{2}, b_{2}, k, p) are compatible.

From the Theorems 1.1 and 1.2 it follows

Corollary 1.1. If l_{i} ∈ \overline{F}_{f} and the functions a_{i}, b_{i} (i = 1, 2) fulfil the condition (1.17), then the tangency relations T_{l_{1}}(a_{1}, b_{1}, k, p) and T_{l_{2}}(a_{2}, b_{2}, k, p) are compatible in the classes of sets M̃_{p,k} ∩ D_{p}(E, l).

2. On the compatibility of the tangency relations of sets of the classes A_{p,k}^{∗}

Let (E, ρ) be a metric space. Let us put by definition (see [3])

A_{p,k}^{∗} = { A ∈ E_{0} : p ∈ A′ and there exists a number λ > 0 such that

lim sup_{[A,p;k]∋(x,y)→(p,p)} \frac{ρ(x, y) − λρ(x, A)}{ρ^{k}(x, p)} ⩽ 0 \}, (2.1)

where

[A, p; k] = { (x, y) : x ∈ E, y ∈ A and ρ(x, A) < ρ^{k}(x, p) = ρ^{k}(y, p) }.  (2.2)

In the paper [4] it was proved that A_{p,k}^{∗} ⊂ M̃_{p,k} for any k > 0 and p ∈ E. With this connection the Theorems 1.1, 1.2 mentioned in Section 1 of this paper are fulfilled in the classes of sets A_{p,k}^{∗} ∩ D_{p}(E, l). It appears that these theorems will be true for sets of the classes A_{p,k}^{∗} ∩ D_{p}(E, l) at slightly weaker conditions concerning the functions a, b, a_{i}, b_{i} (i = 1, 2) appearing in the assumptions of the above theorems.

In the paper [3] the following lemma was proved:

Lemma 2.1. If

\frac{a(r)}{r^{k}} ⟶_{r→0+} 0,  (2.3)

then for an arbitrary set A ∈ A_{p,k}^{∗} ∩ D_{p}(E, ρ)

\frac{1}{r^{k}} d_{ρ}(A ∩ S_{ρ}(p, r)_{a(r)}) ⟶_{r→0+} 0.  (2.4)

From this lemma and from (6) it follows that

\frac{1}{r^{k}} d_{l}(A ∩ S_{l}(p, r)_{a(r)}) ⟶_{r→0+} 0,  (2.5)

for l ∈ \overline{F}_{f} and A ∈ A_{p,k}^{∗} ∩ D_{p}(E, l), when the function a fulfils the condition (2.3).

Similarly as in the case of the classes of sets M̃_{p,k}, using (2.5) we can prove the following theorem.

Theorem 2.1. If l_{i} ∈ \overline{F}_{f} (i = 1, 2),
\frac{a(r)}{r^{k}} ⟶_{r→0+} 0, and \frac{b(r)}{r^{k}} ⟶_{r→0+} 0,  (2.6)

then for arbitrary sets of the classes A_{p,k}^{∗} ∩ D_{p}(E, l) the tangency relations T_{l_{1}}(a, b, k, p) and T_{l_{2}}(a, b, k, p) are compatible.

Let a_{i}, b_{i} (i = 1, 2) be non-negative real functions defined in a certain righthand side neighbourhood of 0 and fulfilling the condition (1.16). In the paper [8] the following theorem was proved.

Theorem 2.2 If l ∈ \overline{F}_{f} and
\frac{a_{i}(r)}{r^{k}} ⟶_{r→0+} 0 and \frac{b_{i}(r)}{r^{k}} ⟶_{r→0+} 0 for i = 1, 2,  (2.7)

then for arbitrary sets of the classes A_{p,k}^{∗} ∩ D_{p}(E, l) the tangency relations T_{l}(a_{1}, b_{1}, k, p) and T_{l}(a_{2}, b_{2}, k, p) are compatible.

From the Theorems 2.1 and 2.2 the following corollary results.

Corollary 2.1. If the functions a_{i}, b_{i} (i = 1, 2) fulfil the condition (2.7) and l_{i} ∈ \overline{F}_{f}, then the tangency relations T_{l_{1}}(a_{1}, b_{1}, k, p) and T_{l_{2}}(a_{2}, b_{2}, k, p) are compatible in the classes of sets A_{p,k}^{∗} ∩ D_{p}(E, l).

Let id denotes the identity function defined in a right-hand side neighbourhood of 0. If we put f = id, then the class \overline{F}_{id} of the function l is equal to the class F_{ρ}^{∗} (see [3], [4]). From here it results that all theorems about the problem of the compatibility of the tangency relations of sets for the functions of the class F_{ρ}^{∗} given in the papers [3] and [4] follow from the theorems of this paper.

REFERENCES

[1] A. Cha̧dzyńska, On some classes of sets related to the symmetry of the tangency relation in a metric space, Ann. Soc. Math. Polon., Comm. Math. 16 (1972), 219–228.

[2] S. Gola̧b et Z. Moszner, Sur le contact des courbes dans les espaces metriques generaux, Colloq. Math. 10 (1963), 105–311.

[3] T. Konik, On the compatibility of the tangency relations of sets of the classes A_{p,k}^{∗} in generalized metric spaces, Demonstratio Math. 19 (1986), 203–220.

[4] T. Konik, On the tangency of sets of some class in generalized metric spaces, Demonstratio Math. 22 (1989), 1093–1107.

[5] T. Konik, On the tangency of sets in generalized metric spaces for certain functions of the class F_{ρ}^{∗}, Mat. vesnik 43 (1991), 1–10.

[6] T. Konik, On the tangency of sets of the class M̃_{p,k}, Publ. Math. Debrecen, 43/3–4 (1993), 329–336.

[7] T. Konik, Tangency relations for sets in some classes in generalized metric spaces, Math. Slowaca (in print).

[8] T. Konik, On the compatibility and the equivalence of the tangency relations of sets of the classes A_{p,k}^{∗}, Journal of Geometry (Basel) (in print).

[9] W. Waliszewski, On the tangency of sets in generalized metric spaces, Ann. Polon. Math. 28 (1973), 275–284.

(received 10.09.1997.)

Institute of Mathematics & Computer Science, Technical University, Da̧browskiego 73, 42-200 Czȩstochowa, Poland