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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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
DOI: https://doi.org/10.1090/S0002-9947-1985-0808739-7
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