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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Winter 2005 • Question 6

Question

Math 6510 • Differentiable Manifolds Winter 2005 • Question 6

Let \(X\) be the standard “middle thirds” Cantor set in \(\mathbb {R}\), viewed as a subset of the plane \(\mathbb {R}^2\). Assume that \(f:X\to \mathbb {R}\) is a function with the property that for every \(x\in X\) there is an open set \(U_x\ni x\) in \(\mathbb {R}^2\) and a smooth function \(f_x:U_x\to \mathbb {R}\) such that \(f_x|_{U_x\cap X}=f|_{U_x\cap X}\). Show that there is an open set \(U\) in \(\mathbb {R}^2\), \(U\supset X\), and a smooth function \(g:U\to \mathbb {R}\) such that \(g|_X=f\).

Raw LaTeX

Let \(X\) be the standard ``middle thirds'' Cantor set in \(\mathbb{R}\), viewed as a
subset of the plane \(\mathbb{R}^2\). Assume that \(f:X\to\mathbb{R}\) is a function with
the property that for every \(x\in X\) there is an open set \(U_x\ni x\) in \(\mathbb{R}^2\)
and a smooth function \(f_x:U_x\to\mathbb{R}\) such that
\(f_x|_{U_x\cap X}=f|_{U_x\cap X}\). Show that there is an open set \(U\) in
\(\mathbb{R}^2\), \(U\supset X\), and a smooth function \(g:U\to\mathbb{R}\) such that
\(g|_X=f\).

Topic Tags

Smooth Extension and Approximation Theorems