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If $B \subset \mathbb{R}^m$ is compact and $x \in \mathbb{R}^n$, it is easy to see that $\{ x \} \times B \subset \mathbb{R}^{m+n}$ is compact.
I would like to know how to prove this.
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Björn Friedrich
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Ponzu
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If you want to rigorously prove this, just use the definition of compactness. Consider a cover, and show that there is a finite subcover. – Euler....IS_ALIVE May 22 '17 at 13:41
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${x} \times B$ is a topological space (with the subspace topology). What are the open sets? – preferred_anon May 22 '17 at 13:45
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2Or just consider the obvious homeomorphism $b\mapsto (x,b).$ – Matematleta May 22 '17 at 13:48