Measure, Integration & Real Analysis · Problem 5C.11

Banach Spaces — Sheldon Axler

Mathematics — Measure Theory · projection-of-sets

Problem

Suppose EE is a subset of Rm×Rn\mathbf{R}^m \times \mathbf{R}^n and A={x∈Rm:(x,y)∈E for some y∈Rn}.A = \{x \in \mathbf{R}^m : (x, y) \in E \text{ for some } y \in \mathbf{R}^n\}. (a) Prove that if EE is an open subset of Rm×Rn\mathbf{R}^m \times \mathbf{R}^n, then AA is an open subset of Rm\mathbf{R}^m. (b) Prove or give a counterexample: If EE is a closed subset of Rm×Rn\mathbf{R}^m \times \mathbf{R}^n, then AA is a closed subset of Rm\mathbf{R}^m.

Browse the problem library