Motivated by constructions in topological data analysis and algebraic combinatorics, we study homotopy theory on the category of Čech closure spaces Cl, the category whose objects are sets endowed with a Čech closure operator and whose morphisms are the continuous maps between them. We introduce new classes of Čech closure structures on metric spaces, graphs, and simplicial complexes, and we show how each of these cases gives rise to an interesting homotopy theory. In particular, we show that there exists a natural family of Čech closure structures on metric spaces which produces a non-trivial homotopy theory for finite metric spaces, i.e. point clouds, the spaces of interest in topological data analysis. We then give a Čech closure structure to graphs and simplicial complexes which may be used to construct a new combinatorial (as opposed to topological) homotopy theory for each skeleton of those spaces. We further show that there is a Seifert-van Kampen theorem for closure spaces, a well-defined notion of persistent homotopy, and an associated interleaving distance. As an illustration of the difference with the topological setting, we calculate the fundamental group for the circle, ‘circular graphs’, and the wedge of circles endowed with different closure structures. Finally, we produce a continuous map from the topological circle to ‘circular graphs’ which, given the appropriate closure structures, induces an isomorphism on the fundamental groups.

受拓扑数据分析和代数组合的构造的影响，我们研究了Čech封闭空间类别Cl的同伦理论，其对象集具有Čech闭包运算符的类别，其射态是它们之间的连续映射。我们在度量空间，图和简单复数上引入了新的Čech闭包结构类，并且我们展示了这些情况中的每一个如何引起有趣的同伦理论。特别是，我们证明了在度量空间上存在一个自然的Čech封闭结构族，该族为有限度量空间（即点云，拓扑数据分析中关注的空间）产生了非平凡的同伦理论。然后，我们为图和简单复合物提供一个Čech封闭结构，可用于为这些空间的每个骨架构造一个新的组合（相对于拓扑）同构理论。我们进一步证明，对于封闭空间，存在一个Seifert-van Kampen定理，一个明确定义的持久同态概念以及相关的交织距离。为了说明与拓扑设置的不同，我们计算了圆的基本组，“圆图”以及具有不同闭合结构的圆的楔形。最后，我们给出了从拓扑圆到“圆图”的连续图，如果有适当的封闭结构，则可以在基团上引起同构。