Bounded Domains and Unbounded Domains

Journal of Formalized Mathematics
Volume 11, 1999
University of Bialystok
Copyright (c) 1999 Association of Mizar Users

Bounded Domains and Unbounded Domains

Yatsuka Nakamura: Shinshu University, Nagano
Andrzej Trybulec: University of Bialystok
Czeslaw Bylinski: University of Bialystok

Summary.

First, notions of inside components and outside components are introduced for any subset of $n$-dimensional Euclid space. Next, notions of the bounded domain and the unbounded domain are defined using the above components. If the dimension is larger than 1, and if a subset is bounded, a unbounded domain of the subset coincides with an outside component (which is unique) of the subset. For a sphere in $n$-dimensional space, the similar fact is true for a bounded domain. In 2 dimensional space, any rectangle also has such property. We discussed relations between the Jordan property and the concept of boundary, which are necessary to find points in domains near a curve. In the last part, we gave the sufficient criterion for belonging to the left component of some clockwise oriented finite sequences.

MML Identifier: JORDAN2C

The terminology and notation used in this paper have been introduced in the following articles [38] [9] [45] [32] [46] [7] [8] [3] [40] [18] [2] [1] [34] [47] [13] [20] [6] [31] [33] [17] [29] [36] [15] [4] [10] [44] [41] [35] [5] [21] [30] [37] [24] [11] [14] [26] [12] [43] [42] [16] [19] [22] [27] [23] [28] [39] [25]

Contents (PDF format)

Definitions of Bounded Domain and Unbounded Domain
Bounded and Unbounded Domains of Rectangles
Jordan Property and Boundary Property
Points in LeftComp

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Received January 7, 1999

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