Measure, Integration & Real Analysis · Problem 5C.9

Banach Spaces — Sheldon Axler

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

Problem

Suppose

G_1

is a nonempty subset of

\mathbf{R}^m

and

G_2

is a nonempty subset of

\mathbf{R}^n

. Prove that

G_1 \times G_2

is an open subset of

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

if and only if

G_1

is an open subset of

\mathbf{R}^m

and

G_2

is an open subset of

\mathbf{R}^n

. [One direction of this result was already proved (see 5.36); both directions are stated here to make the result look prettier and to be comparable to the next exercise, where neither direction has been proved.]

Related theory

Metric Spaces Open And Closed Sets Convergence

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