Let X be a closed subspace of a metric space M. It is well known that, under mild hypotheses, one can estimate the Betti numbers of X from a finite set \(P\subset M\) of points approximating X. In this paper, we show that one can also use P to estimate much more detailed topological properties of X. We achieve this by proving the stability of \(A_\infty \)-persistent homology. In its most general case, this stability means that given a continuous function \(f:Y\rightarrow {\mathbb {R}}\) on a topological space Y, small perturbations in the function f imply at most small perturbations in the family of \(A_\infty \)-barcodes. This work can be viewed as a proof of the stability of cup-product and generalized-Massey-products persistence. The technical key of this paper consists of figuring out a setting which makes \(A_\infty \)-persistence functorial.

令X是度量空间M的闭子空间。众所周知，在温和的假设下，可以从逼近X的点 的有限集合\(P\subset M\)估计X的 Betti 数。在本文中，我们表明还可以使用P来估计X 的更详细的拓扑属性 。我们通过证明\(A_\infty \) -持久同源性的稳定性来实现这一点。在最一般的情况下，这种稳定性意味着在拓扑空间Y上 给定一个连续函数\(f:Y\rightarrow {\mathbb {R}}\)，函数f中的小扰动暗示在\(A_\infty \) -barcodes家族中最多有小扰动。这项工作可以看作是杯积和广义梅西积持久性稳定性的证明。本文的技术关键在于找出一个设置，使\(A_\infty \) -persistence 函子化。