0
$\begingroup$

This statement to this effect is in a set of notes on topology:

With $X$ and $Y$ metric spaces and $X$ is unbounded, $X\times Y$ is bounded if $Y$ is empty.

I would appreciate help to show this.

I know that $d_{X\times Y} = d_{X} \times d_{Y}$ where the $d$'s are the respective metrics.

(If I understood them correctly, I saw in some posted answers that a metric on the $\emptyset$ is -$\infty$. But even if this is correct, I don't see how it would help.)

1 Answers 1

2

If $Y$ is empty, then $X \times Y$ is also empty. An empty metric space is bounded.

  • 0
    $+1$ of course. I was wondering if this could naively be approached using the formula for the metric of a product space I wrote above? Thanks. Regards, – 2017-01-18
  • 0
    @TheBirdistheWord Whatever your metric happens to be, an empty metric space will be vacuously bounded. You could technically say that we're using your formula, in a trivial sense. – 2017-01-18