In this paper, we present a new approach to the Mandelbrot set via fuzzy set theory. We introduce and characterize the fuzzy Mandelbrot set by describing its membership function that determines the degrees of belonging of the points to this set under the iteration even if the orbit of the point is not limited. Then we investigate the basic topological properties of the fuzzy Mandelbrot set and its strong α-cut such as closedness, openness, boundness, connectedness and simply connectedness.

中文翻译：

---

模糊 Mandelbrot 集

在本文中，我们提出了一种通过模糊集理论处理 Mandelbrot 集的新方法。我们通过描述它的隶属函数来引入和表征模糊Mandelbrot集，该隶属函数决定了点在迭代下对该集合的归属程度，即使该点的轨道不受限制。然后我们研究了模糊Mandelbrot集的基本拓扑性质及其强α-割，如封闭性、开放性、有界性、连通性和简单连通性。