Every compactum $K\subset \widehat{\mathbb{C}}$ has a core decomposition ${\mathcal{D}}_K^{PC}$ with Peano quotient, which is the finest one among all upper semi-continuous decompositions into sub-continua such that the quotient space is a Peano compactum [14]. The properties of core decomposition can provide information about the topology of K. Each member of ${\mathcal{D}}_K^{PC}$ is called an atom of K. In this paper, it is shown that if {x} is an atom of a continuum $K\subset \widehat{\mathbb{C}}$ then K is locally connected at x; however, the converse is not true. Moreover, we will obtain singleton atoms of the Mandelbrot set $\mathcal{M},$ based on well known points at which $\mathcal{M}$ is locally connected.

中文翻译：

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关于Mandelbrot集的核心分解的注释

每个紧凑 $ķ\subset \widehat{\mathbb{C}}$ 有核心分解 ${\mathcal{d}}_{ķ}^{PC}$Peano商，它是所有上半连续分解成亚连续体中最好的一个，因此商空间是Peano compactum [14]。核心分解的性质可以提供有关K拓扑的信息。每个成员${\mathcal{d}}_{ķ}^{PC}$被称为K的原子。在本文中，它被示出，如果{ X }是一个连续的原子$ķ\subset \widehat{\mathbb{C}}$那么K在x本地连接；但是，事实并非如此。此外，我们将获得Mandelbrot集的单子原子$\mathcal{中号}，$ 根据众所周知的观点 $\mathcal{中号}$ 已本地连接。