A shape theoretical approach to fractal geometry

形状理论是关于具有局部不良行为的拓扑空间的同伦理论。另一方面，分形是带有距离概念的空间之一，被认为是具有局部复杂结构的度量空间。在这个项目中，我们应用形状理论，特别是它的范畴方面，来寻找分数维，甚至一般度量空间的性质。特别地，我们证明了逆系的概念和正态序列的概念与度量空间有很强的联系，逆系是形式论最重要的方法之一，正态序列是拓扑空间中最重要的概念之一。

Shape theory is known as a homotopy theory for topological spaces with locally bad behavior. Fractal, on the other hand, is one of the spaces carrying the notion of distance, and is known as a metric space with locally complicated structures. In this project, we apply the theory of shape, in particular, its categorical aspects, to find the properties of fractals, or even general metric spaces. In particular, we have shown that the notion of inverse system, which is one of the most important approaches to shape theory, and the notion of normal sequence, which is one of the most important notions in the theory of topological spaces, have strong connections to metric spaces.