Cantorkészletre

Cantorkészletre is a Hungarian term that translates to "Cantor set" in English. The Cantor set is a fractal, a set of points that exhibits self-similarity on all scales. It is constructed by starting with the closed interval [0, 1] and iteratively removing the middle third of each remaining segment.

The construction begins with the interval [0, 1]. In the first step, the open middle third (1/3, 2/3) is removed, leaving two closed intervals: [0, 1/3] and [2/3, 1]. In the second step, the middle third of each of these remaining intervals is removed, resulting in four closed intervals: [0, 1/9], [2/9, 1/3], [2/3, 7/9], and [8/9, 1]. This process is continued infinitely.

The Cantor set has several paradoxical properties. Despite being constructed by removing intervals, it is an uncountable set, meaning it contains as many points as the original interval [0, 1]. It has a Lebesgue measure of zero, indicating that it occupies no space in the traditional sense. Furthermore, every point in the Cantor set is an isolated point, meaning no point is an endpoint of any interval remaining in its construction. The Cantor set serves as a fundamental example in topology and real analysis, illustrating concepts of measure theory, topology, and the nature of infinite sets.