The tangency of sets and monotonicity function

(1)

THETANGENCYOF SETS AND MONOTONICITY FUNCTION

Jerzy Grochulski

Institute of Mathematics and Computer Science, Czestochowa University of Technology

Abstract. The present paper deals with the connections between tangency relations of sets T_l(a_i,b_i,k,p) (l = 1,2) in the metric space (E,ρ) and monotonicity function l.

Let (E,ρ) be a metric space. Let the set

(

^p ^t

) {

^x ^E

(

^p ^x

)

^t

}

S , = ∈ :ρ , = (1)

denote the sphere with centre of the point p and the radius t, and

(

p t

) {

x E

(

p x

)

t

}

K , = ∈ :ρ , < (2)

denote the ball with centre of the point p and the radius t.The a set

( )

( )K

( )

q t r

p S

r p S

t q ,

,

∈ ,

= U for t > 0 (3)

and

(

^p ^r

)

^S

(

^p ^t

)

S , _t = , for t = 0 (4)

will be called the t - neighbourhood of the sphere. The E₀ be the family of all non- -empty subset of set E.

Let a and b are non-negative real functions defined in a right-hand neighbourhood of the point 0 such that

( )

r ^^r→^₀+^→⁰

a and a

( )

r ^^r→^₀+^→⁰ (5) The pair (A,B) is (a,b) - concentrated at the point p if 0 is a concentration point of the set all real numbers r > 0 such that the sets A∩S

(

p,r

)

_a_{( )}_r and B∩S

(

p,r

)

_b_{( )}_r

are non-empty.

Let l be a real non-negative function defined on the Cartesian product E₀xE₀ satisfying the condition

{ } { }

(

x y

) (

x y

)

l , =ρ , for x, y∈E (6)

(2)

The set A is (a,b) - tangent order k at the point p ∈ E to the set B if the pair (A,B) is (a,b) - concentrated at the point p and

(

⁽ ^, ⁾ ^, ⁽ ^, ⁾

)

⁰

1

) 0 ( )

( ∩ →

∩

→ + r r b r

kl A S p r a B S p r

r (7)

where k is an arbitrary positive real number. The relation

( )

( ) ( )







 ∈ ∧ ∩ ∩  →

=

→ + 0

) , ( , ) , 1 (

, : ,

, , ,

) 0 ( )

(

0 k a r br r

l

r p S B r p S A r l E B A B A

p k b a T

(8)

will be called the relations of tangency of the set in the metric space _(E,ρ). THEOREM 9. If 0 ≤ t1 ≤ t2, t₁,t₂ ∈ R (the R is set of real numbers), then

( ) ( )

2

1 ,

,r _t S p r _t p

S ⊂

Proof. Let t₁> 0 then and

( )

, 1

r t

p

x ∈S from (3) exists a point q∈S

(

p,r

)

such that

(

q^,t₁

)

^,

K

x ∈ thereρ

(

q^,x

)

<t₁≤t₂ and so x ∈K

(

p^,t₂

)

to say

(

^,

)

^.

t2

t q

x ∈S Let t₁= 0 then

(

^,

) (

^,

)

^.

1 S p r

r p

S _t = Let x∈S

(

p,r

)

to say ρ

(

p,x

)

=r and x ∈K

(

p^,t₂

)

^, therefore

(

^,

)

^.

t2

t q

x ∈S This ends the proof.

THEOREM 10. If

(

A^,B

)

l

(

C^,B

)

l ≤ for A⊂ C

(

^A^,^B^,^C∈^E₀

)

(11) and a₁

( )

r ≤a₂

( )

r for r > 0, then

(

A,B

)

∈T₁

(

a₂,b,k,p

)

⇒

(

A,B

)

∈T₁

(

a₁,b,k,p

)

Proof. Let

(

A,B

)

∈T₁

(

a₂,b,k,p

)

and a₁

( )

r ≤a₂

( )

r for r > 0. Then ( ) 0

1 0 →

→ + r

r

a

from here and from theorem (9)S

(

p r

)

_a_{( )}_r S

(

p r

)

_a _{( )}_r

2

1 ,

, ⊂ for say

(

( ) ( )

)

1

(

( , ) ( ), ( , ) ( )

)

) , ( , ) , 1 (

2 2

1r a r k a r br

k a l A S p r B S p r

r r p S B r p S A

r l ∩ ∩ ≤ ∩ ∩

from here

(

A^,B

)

∈T₁

(

a₁^,b^,k^,p

)

^. This ends the proof.

(3)

Example 12. The function

(

^A ^B

) {

^diam

(

^x ^B

)

^x ^A

}

l₅ , =sup { }∪ ; ∈

by diamA we shall denote the diameter of the set A, the A,B∈E₀,satisfying the condition (6) and (11), therefore for

(

A,B

)

∈T₁

(

a₁,b,k,p

)

⇒

(

A,B

)

∈T₁

(

a₂,b,k,p

)

THEOREM 13. If

(

^A^,^B

) (

^l ^A^,^D

)

l ≤ for B⊂D

(

A^,B^,D∈E₀

)

(14) and ^b₁

( )

^r ≤^b₂

( )

^r for r > 0, then

(

A,B

)

∈T₁

(

a,b₁,k,p

)

⇐

(

A,B

)

∈T₁

(

a,b₂,k,p

)

Proof. Let

(

A,B

)

∈T₁

(

a,b₂,k,p

)

and ^b₁

( )

^r ≤^b₂

( )

^r for r > 0. Then ( ) 0

1 0 →

→ + r

r

b

from here and from theorem (9) S

(

p r

)

_b_{( )}_r S

(

p r

)

_b _{( )}_r

2

1 ,

, ⊂ for say

(

( ) ₁( )

)

1

(

( , ) ( ), ( , )₂( )

)

) , ( , ) , 1 (

r b r

k a r b r

r r p S B r p S A

r l ∩ ∩ ≤ ∩ ∩

from here

(

A^,B

)

∈T₁

(

a^,b₂^,k^,p

)

^. This ends the proof.

Example 15. The function l₇ by the formula

(

^A ^B

)

^diam

(

^A ^B

)

l₇ , = ∪

satisfying the condition (6) and (14), therefore fora₁

( )

r ≤a₂

( )

r as r > 0

(

A,B

)

∈T₁

(

a,b₁,k,p

)

⇐

(

A,B

)

∈T₁

(

a,b₂,k,p

)

THEOREM 16. If

(

A B

)

l

(

C B

)

l , ≥ , for ^A⊂^C

(

^A^,^B^,^C∈^E₀

)

(17) and a₁

( )

r ≤a₂

( )

r for r > 0 and

( )

⁰^,

2 0 →

→ + r

r

a then

(

^A^,^B

)

^∈^T₁

(

^a₂^,^b^,^k^,^p

)

^⇒

(

^A^,^B

)

^∈^T₁

(

^a₁^,^b^,^k^,^p

)

(4)

Proof. Let

(

A,B

)

∈T₁

(

a₂,b,k,p

)

and a₁

( )

r ≤a₂

( )

r for r > 0. Then

( )

⁰

1 0 →

→ + r

r a from here and from theorem (9) S

(

p r

)

_a _{( )}_r S

(

p r

)

_a _{( )}_r

2

1 ,

, ⊂ for say

(

( ) ( )

)

1

(

( , ) ( ), ( , )( )

)

) , ( , ) , 1 (

1 2

1r a r k a r b r

r r p S B r p S A

r l ∩ ∩ ≤ ∩ ∩

from here

(

A^,B

)

∈T₁

(

a₂^,b^,k^,p

)

^.This ends the proof.

Example 18. The function l₆ by the formula

(

^A ^B

) {

^diam

( { }

^x ^B

)

^x ^A

}

l₆ , =inf ∪ ; ∈

satisfying the condition (6) and (14), therefore for a₁(r)≤a₂(r) as r > 0

(

A,B

)

∈T₁

(

a,b₁,k,p

)

⇒

(

A,B

)

∈T₁

(

a,b₂,k,p

)

THEOREM 19. If

(

^A ^B

) (

^l ^A ^D

)

l , ≥ , for B⊂D

(

A,B,D∈E₀

)

(20) and ^b₁

( )

^r ≤^b₂

( )

^r for r > 0 and

( )

0,

2 0 →

→ + r

r

b then

(

A,B

)

∈T₁

(

a,b₁,k,p

)

⇒

(

A,B

)

∈T₁

(

a,b₂,k,p

)

Proof. Let

(

A,B

)

∈T₁

(

a,b₁,k,p

)

and a₁

( )

r ≤a₂

( )

r for r > 0. Then from here and from theorem (9)S

(

p r

)

_a_{( )}_r S

(

p r

)

_a _{( )}_r

2

1 ,

, ⊂ for say

(

( ) ₂( )

)

1

(

( , ) ( ), ( , )₁( )

)

) , ( , ) , 1 (

r b r

k a r b r

r r p S B r p S A

r l ∩ ∩ ≤ ∩ ∩

from here

(

A^,B

)

∈T₁

(

a₂^,b^,k^,p

)

^.This ends the proof.

The metric ρ induces some l_i(x) defined by formulas:

} };

);

, ( inf{inf{

) ,

1(A B x y y B x A

l = ρ ∈ ∈

} };

);

, ( sup{inf{

) ,

2(A B x y y B x A

l = ρ ∈ ∈

} };

);

, ( inf{sup{

) ,

3(A B x y y B x A

l = ρ ∈ ∈

} };

);

, ( sup{sup{

) ,

4(A B x y y B x A

l = ρ ∈ ∈

}

; } { sup{

) ,

5(A B diam x B x A

l = ∪ ∈

} );

} ({

inf{

) ,

6(A B diam x B x A

l = ∪ ∈

(

^A ^B

)

diam B

A

l₇( , )= ∪

(5)

The function l₁ satisfying conditions (17) and (19). The function l₂ satisfying condition (11) and (14). The function l₃ satisfying condition (17) and (14). The function l₄ satisfying condition (11) and (14).

THEOREM 21. For A,B∈E₀

(

A B

) {

diamA diamBl

(

A B

) }

l₈ , =max , , ₄ ,

Proof. Let diam

(

A∪B

)

=s<∞^. Let theε >0, exist point x,y∈A∪B such as this

(

x^,y

)

s^.

s−ε <ρ ≤ Then at least one inequality:

(

^x ^y

)

^s

s−ε<ρ , ≤ for x,y∈A

(

x y

)

s

s−ε<ρ , ≤ for x,y∈B s−ε<ρ

(

x,y

)

≤s for x∈ ,A y∈B is true.

From here diam

(

A∪B

)

=diamA and diamB ≤diamA and ^l₄

(

^A,^B

)

≤^diamA or diam

(

A∪B

)

=diamB and diamA ≤diamB and ^l₄

(

^A,^B

)

≤^diamB else ^diam

(

^A∪^B

)

=^l₄

(

^A^,^B

)

and ^diamA≤^l₄

(

^A^,^B

)

and ^diamB≤^l₄

(

^A^,^B

)

therefore

(

A B

) {

diamA diamBl

(

A B

) }

diam ∪ = <

, 4

, max

Let diam

(

A∪ B

)

=∞^, then exist point x,y∈A∪B that such ρ

(

^x^,^y

)

>^N^, where N is arbitrary positive real number, then at least one inequality: ρ

(

x,y

)

>N for

, ,y A

x ∈ ρ

(

x,y

)

>N for x,y∈B, ρ

(

x,y

)

>N for x∈A,y∈B. From here

∞

diamA= or diamB=∞ else l₄

(

A,B

)

=∞. This ends the proof.

THEOREM 22. For A,B∈E₀ ^l₉

(

^A^,^B

)

=^max

{

^diam

(

^A^,^l₃

(

^A^,^B

) ) }

^.

Proof. Let the _{A =}

{ }

^x for a x ∈ to sayE diamA=0.Then from Theorem (21)

{ }

(

x B

) {

diamB

{ (

x y

)

y B

} }

diam ∪ =max ,sup ρ , ; ∈

Therefore

(

A B

) { {

diamB

{ (

x y

)

y B

} }

x A

}

l₇ , =inf max ,sup ρ , ; ∈ ; ∈ Let

( )

{ }

{

diamB,inf sup x,y;y∈B;x∈A

}

=diamB

max ρ

that is

( )

{ }

{

sup x,y;y∈B;x∈

}

≤diamB

inf ρ

(6)

to say exist a point x ∈₀ A such that sup

{

ρ

(

x₀,y

)

;y∈B

}

≤diamB. For an arbitrary point x ∈ A

( )

{ }

( )

{ }

{

diamB x y y B

}

diamB A x B y y x diamB

diamB

=

∈

≤

∈

∉

≤

; , sup , max

;

; , sup , max

inf

ρ 0

ρ

Let

( )

{ }

{

diamB,inf sup x,y ;y∈B;x∈A

}

=inf

{

sup

{ (

x,y

)

;y∈B

}

;x∈A

}

max ρ ρ

then

( )

{ }

{

x y y∈B x∈A

}

≥diamB

≥ sup , ; ;

inf ρ

that is

( )

{

x,y;y∈B

}

≥diamB

sup ρ for x ∈ A

For arbitrary point x ∈ A

( )

{

x,y ;y∈B

}

≤max

{

diamB,sup

{ (

x,y

)

;y∈B

} }

sup ρ ρ

from here

( )

{ }

{ } { { { ( ) } } }

( )

{ }

{

diamB x y y B

} { (

x y

)

y B

}

A x B y y x diamB

A x B y y x

∈

≤

∈

≤

∈

=

∈

; , sup

; , sup , max

;

; , sup , max

inf

;

; , sup inf

ρ ρ

Therefore

( )

{ }

{

sup x,y;y∈B;∈A

}

=inf

{

max

{

diamB,sup

{ (

x,y

)

;y∈B

} }

;x∈A

}

inf ρ ρ

This ends the proof.

References

[1] Gołąb S., Moszner Z., Sur le contact des courbes dans les espaces metrriques generaux, Colloq.

Math. 1963, 10, 305-311.

[2] Grochulski J., Konik T., Tkacz M., On the equivalence of certain relations of arcs in metric spaces, Demonstratio Math. 1978, 11, 261-271.

[3] Grochulski J., Some properties of tangency relations, Demonstratio Math. 1995, 28, 361-367.

[4] Grochulski J., Some connections between tangency relations, SIMI 2000, Wydawnictwo Politech- niki Częstochowskiej, Częstochowa 2000, 27-32.

[5] Waliszewski W., On the tangency of sets in generalized metric spaces, Ann. Polon. Math. 1973, 28, 275-284.