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Joint continuity of division of smooth functions. II. The distance to a Whitney stratified set from a transversal submanifold

Author: Mark Alan Mostow
Journal: Trans. Amer. Math. Soc. 292 (1985), 585-594
MSC: Primary 58C25
MathSciNet review: 808739
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Abstract: Let $ S$ be a closed set in $ {{\mathbf{R}}^m}$, and let a $ {C^1}$ Whitney stratification of $ S$ be given. (Actually, only Whitney (a)-regularity is needed.) Let $ f:{{\mathbf{R}}^n} \to {{\mathbf{R}}^m}$ be a $ {C^1}$ map transversal to all the strata. Assume that the image of $ f$ intersects $ S$. Then for each compact set $ K$ in $ {{\mathbf{R}}^n}$, the Euclidean distances $ \rho (x,{f^{ - 1}}(S))$ and $ \rho (f(x),S)$, for $ x$ in $ K$, are bounded by constant multiples of each other. The bounding constants can be chosen to work for all maps $ g$ which are close enough to $ f$ in a $ {C^1}$ sense on a neighborhood of $ K$. This result is used in part I (written jointly with S. Shnider) to prove a result on the joint continuity of the division of smooth functions [MS].

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Keywords: Whitney stratification, transversality, Whitney (a)-regularity, regular separation
Article copyright: © Copyright 1985 American Mathematical Society

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