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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2005 • Question 6

Question

Math 6510 • Differentiable Manifolds Fall 2005 • Question 6

Prove that for any two disjoint closed subsets \(A,B\subset \mathbb {R}^n\) there is a smooth function \(f:\mathbb {R}^n\to \mathbb {R}\) such that \(f=0\) on \(A\) and \(f=1\) on \(B\).

Raw LaTeX

Prove that for any two disjoint closed subsets \(A,B\subset\mathbb{R}^n\) there is
a smooth function \(f:\mathbb{R}^n\to\mathbb{R}\) such that \(f=0\) on \(A\) and \(f=1\) on
\(B\).

Topic Tags

Point-Set Topology and Separation Theorems Smooth Extension and Approximation Theorems