Unstable interior points in attainable sets: cracks

Let X be a subset of a linear space. We refer to a continuous function g: X→Rⁿ as an open covering of y₀εRⁿ, respectively of A⊂Rⁿ, if g(X) contains a neighborhood of y₀, respectively of A. The main subject of the paper deals with the sufficient conditions for the instability of a function g as an open covering. Let X be a metric space, Y a normed vector space, g: X→Y continuous, and C⊂g(X). We refer to C as a crack of g if, for every continuous function ω: X→[0, 1] that is positive on g^-1(C), there exists a continuous perturbation e: X→Y such that |e(x)|≤ω(x) for all xεX and (g+e)(X)∪C=O. Thus the presence of a crack in the interior of g(X) reveals the instability of g as an open covering. Subject to certain "qualitative" assumptions, we derive conditions equivalent, to the statement that C is a crack of g. Each of the first two conditions, which deal with the case that C is an oriented C¹ manifold without boundary of dimension n-1, is essentially equivalent to the statement that every topological component Γ of g^-1(C) has a neighborhood Σ^Γ that is mapped by g into one side of C only. The third condition deals with the case that C is a set whose components are C¹ manifolds without boundary in Rⁿ of codimensions 2 or higher. We also provide examples to illustrate the applications of some of the results.