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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 1997 • Question 17

Question

Math 6510 • Differentiable Manifolds Fall 1997 • Question 17

Let \(X=\{(x,y,z)\in \mathbb {R}^3\mid xyz=0\}\), i.e. \(X\) is the union of the three coordinate planes in \(\mathbb {R}^3\). Let \(0\in X\) be the origin. Prove that any homeomorphism of \(X\) must fix 0.

Raw LaTeX

Let \(X=\{(x,y,z)\in\mathbb{R}^3\mid xyz=0\}\), i.e. \(X\) is the union of the
three coordinate planes in \(\mathbb{R}^3\). Let \(0\in X\) be the origin. Prove that any
homeomorphism of \(X\) must fix 0.

Topic Tags

Smooth Manifolds and Embedded Submanifolds