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calef comments on Approaching Infinity - Less Wrong
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Sure it does -- the extended real line is easily made into a metric space.
(The point being that your use of the term "standard real analysis" here is a bit off, specifically in the form of being insufficiently meta. "Real analysis" is a subject in which one considers such ideas as "metrics" (and many other things) in general; the term doesn't just refer narrowly to the properties of the real line equipped with all the most "standard" structures.)
Fair enough, I was simply trying to appeal to what is probably his most familiar intuition regarding the real number line.
Most people that are going to have confusion about a big number being close to infinity probably aren't going to know what a metric space is.
They may not know the term "metric space" (in which case you just explain that it's a setting where we can measure distance), but if they think larger numbers are "closer to infinity" than smaller numbers, that means they are intuitively thinking in terms of the extended real line (metrized in one of the usual ways).