Positively separated sets
In this article, we will explore the topic of Positively separated sets from different perspectives and approaches. Positively separated sets is a broad and relevant topic that has captured the attention of various sectors of society in recent years. We will address fundamental aspects of Positively separated sets, examining its impact in different areas and its evolution over time. In addition, we will analyze the different points of view that exist around Positively separated sets, as well as the implications it has on people's daily lives. Through this article, we seek to provide a complete and updated view on Positively separated sets, with the aim of offering our readers a deeper and enriching knowledge on this topic.
In mathematics, two non-empty subsets A and B of a given metric space (X, d) are said to be positively separated if the infimum
(Some authors also specify that A and B should be disjoint sets; however, this adds nothing to the definition, since if A and B have some common point p, then d(p, p) = 0, and so the infimum above is clearly 0 in that case.)
For example, on the real line with the usual distance, the open intervals (0, 2) and (3, 4) are positively separated, while (3, 4) and (4, 5) are not. In two dimensions, the graph of y = 1/x for x > 0 and the x-axis are not positively separated.
References
- Rogers, C. A. (1998). Hausdorff measures. Cambridge Mathematical Library (Third ed.). Cambridge: Cambridge University Press. pp. xxx+195. ISBN 0-521-62491-6.
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