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Let $(X,d)$ be a compact metric space of finite diameter. Recall that we can always compute the Hausdorff distance between subsets $A, B \subset X$ via $$ d_H(A,B) = \max\left[\sup_{a\in A}\inf_{b\in …

In the absence of convex images, one typically relies on algebraic topology as you have guessed. If your set-valued map has a reasonably nice domain and contractible images, then you can easily string …

One way to make things "minimal" (given the lack of any further information) is to construct a simplicial complex whose cup products are all trivial, so the (co)homology generators don't interact with …

The following question came up while constructing delay embeddings of time series data. Consider an unknown topological space $X$ and an unknown continuous function $f:X \to X$. We are given a combin …

At the risk of sounding (oxy?)moronic, I'd say that the term "stratification" is locally standard. Meaning, there exist (at least) three communities which agree internally on what the term means, but …

Bad News The answer to Q3 as stated is no. Let $X$ be the Michael line, and let $Y$ be the closed subset consisting of all the rationals. Then, there is no bounded linear extension $C(Y,\mathbb{R}) \t …

Throughout this question, notation of the form $(X,A)$ denotes a sufficiently nice pair of topological spaces. I think for most of what I'm saying here, it is enough to assume that the inclusion $A \h …