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Ok, so we're given the distance, d(x,C) between a point, x, and a closed set C in a metric space to be: inf{d(x,y): for all y in C}. Then we have to generalize this to define the distance between two sets I'm fairly certain you can define it as:

the distance between closed sets D and C in a metric space, d(C,D) = inf{d(y,D): for all y contained in C}. Which should be equivalent to inf{d(x,y): for all x,y contained in C,D respectively}.

My question is this: How to construct an example of two closed, disjoint sets whose distance is zero under this definition? I feel like I need to find two sets containing points that can be made arbitrarily close, but am unsure how to do this without some point being a limit point of both sets, contradicting C,D disjoint.

If you guys have an idea that would be great, I'd much prefer a hint or nudge in the right direction if possible.

Thanks again.

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# Distance Between Closed sets in a metric space

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