Journal of Formalized Mathematics
Volume 2, 1990
University of Bialystok
Copyright (c) 1990 Association of Mizar Users

## Submetric Spaces --- Part I

Technical Univercity of Rzeszow
Mariusz Startek
Technical University of Rzeszow

### Summary.

Definitions of pseudometric space, nonsymmetric metric space, semimetric space and ultrametric space are introduced. We find some relations between these spaces and prove that every ultrametric space is a metric space. We define the relation {\it is between}. Moreover we introduce the notions of the open segment and the closed segment.

Supported by RPBP.III-24.B3.

#### MML Identifier: SUB_METR

The terminology and notation used in this paper have been introduced in the following articles [7] [4] [10] [8] [11] [2] [3] [1] [6] [9] [5]

Contents (PDF format)

#### Bibliography

[1] Czeslaw Bylinski. Binary operations. Journal of Formalized Mathematics, 1, 1989.
[2] Czeslaw Bylinski. Functions and their basic properties. Journal of Formalized Mathematics, 1, 1989.
[3] Czeslaw Bylinski. Functions from a set to a set. Journal of Formalized Mathematics, 1, 1989.
[4] Czeslaw Bylinski. Some basic properties of sets. Journal of Formalized Mathematics, 1, 1989.
[5] Stanislawa Kanas, Adam Lecko, and Mariusz Startek. Metric spaces. Journal of Formalized Mathematics, 2, 1990.
[6] Andrzej Trybulec. Binary operations applied to functions. Journal of Formalized Mathematics, 1, 1989.
[7] Andrzej Trybulec. Tarski Grothendieck set theory. Journal of Formalized Mathematics, Axiomatics, 1989.
[8] Andrzej Trybulec. Subsets of real numbers. Journal of Formalized Mathematics, Addenda, 2003.
[9] Andrzej Trybulec and Czeslaw Bylinski. Some properties of real numbers operations: min, max, square, and square root. Journal of Formalized Mathematics, 1, 1989.
[10] Zinaida Trybulec. Properties of subsets. Journal of Formalized Mathematics, 1, 1989.
[11] Edmund Woronowicz. Relations and their basic properties. Journal of Formalized Mathematics, 1, 1989.