Critical point set is closed: Difference between revisions
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{{regular value fact}} | |||
==Statement== | ==Statement== | ||
Let <math>f:M \to N</math> be a [[smooth map]] of [[differential manifold]]s. Then, the set of [[critical point]]s of <math>f</math> is a [[tps:closed subset|closed subset]] of <math>M</math>; equivalently, the set of [[regular point]]s of <math>f</math> is an [[tps:open subset|open subset]] of <math>M</math>. | Let <math>f:M \to N</math> be a [[smooth map]] of [[differential manifold]]s. Then, the set of [[critical point]]s of <math>f</math> is a [[tps:closed subset|closed subset]] of <math>M</math>; equivalently, the set of [[regular point]]s of <math>f</math> is an [[tps:open subset|open subset]] of <math>M</math>. | ||
Latest revision as of 19:36, 18 May 2008
This article gives the statement and possibly proof of a theorem that discusses regular values, critical values, regular points or critical points of a smooth map between differential manifolds
Statement
Let be a smooth map of differential manifolds. Then, the set of critical points of is a closed subset of ; equivalently, the set of regular points of is an open subset of .