Measure, Integration & Real Analysis · Problem 5C.10

Banach Spaces — Sheldon Axler

Mathematics — Measure Theory · product-of-closed-sets

Problem

Suppose

F_1

is a nonempty subset of

\mathbf{R}^m

and

F_2

is a nonempty subset of

\mathbf{R}^n

. Prove that

F_1 \times F_2

is a closed subset of

\mathbf{R}^m \times \mathbf{R}^n

if and only if

F_1

is a closed subset of

\mathbf{R}^m

and

F_2

is a closed subset of

\mathbf{R}^n