CONSTRUCTION OF FLAT IMAGES IN AN ARBITRARY METRIC SPACE

Introduction. In an arbitrary metric space, the basic geometric objects - a straight line, an angle, a plane, are considered subject to the completeness of this space. Usually, these objects in metric space are considered as continuous mappings of the corresponding classical Euclidean geometry objects. If we use only some of the properties of these objects, then we can consider the corresponding images in space, without the requirement of its completeness. In determining the images of objects, one should use only the concept of the metric of space, without involving the passage to the limit. Purpose. The aim of the paper is to define a flat image for a set of points of an arbitrary metric space, without using the completeness property of this space. The aim, too, is to study the relationship between the straight-line arrangement and the flat arrangement of points in space. Results. In work the following concepts are used. Let a , b , c – arbitrary points of metric space (X , ) . An ordered triple of these points (a,b, c) will be called an angle with a vertex at the b point, and denote (a,b, c) . The angular characteristic of the angle (a,b, c) will be called the real number, which is represented by the formula: 2 ( , ) ( , ) ( , , ) ( , ) ( , ) ( , ) 2 2 2 a b b c a b c a b b c a c          . The metric space, in which the notion of the angle and its characteristics are introduced, will be denoted  . We will say that the points a , b , c are straight-line arrangement in the space  , if equality (a,b, c)  1 is performed. If equality (a,b, c)  1 is performed, then the angle is deployed. A plurality of points of space will be called in a straight-line arrangement or straight-line image if any three points of this set are straight-line arrangement. We will say that the four points a , b , c , d space  are flat arrangement, if equality ( , , ) ( , , ) 1 ( , , ) 1 ( , , ) 1 ( , , ) ( , , )  a b d c b d a b c c b d a b c a b d       is performed. We will say that the set of points of space is flat arrangement, if any four points of this set are flat arrangement. The following main results are obtained. Lemma 1. If the points a , b , c are straight-line arrangement in the space  , they are flat arrangement. Lemma 2. Let the points a , b , c are straight-line arrangement in the space  , and, the angle (a,b, c) is deployed. In order for the points a , b , c , d of this space to be flat arrangement, it is necessary and sufficient that equality (a,b, d)   (c,b, d ) be fulfilled. Theorem 1. In order for the points a ,b , c , d the space  to be flat arrangement, it is necessary and sufficient that equality  (a,b,c) (a,b, d) (c,b, d)  (1  2 (a,b,d ))(1  2 (c,b, d)) be fulfilled . Originality. In this paper the concept of a flat arrangement of points of an arbitrary metric space is introduced for the first time. Necessary and sufficient conditions for such placement for four different points of space are obtained. Conclusion. The flat arrangement of points possesses the basic properties of the classical plane. The concept of a flat allocation of points of a metric space can be used to construct geometric images of classical geometric figures. The work should be continued in the direction of obtaining conditions of perpendicularity and parallelism of the sets of points of an arbitrary metric space.