Find all metrics on a set $X$ consisting of two points. Consisting of one point.

Question

First, this is an exercise in the first section of Kreyszig's introductory functional analysis text. In this section he has already given several examples of metric spaces, including: $l^\infty$, $C[a,b]$, and an example of a discrete metric space. In this section he has stated that we can interpret things like infinite but bounded sequences (for $l^\infty$) as points, or continuous functions on closed intervals (for $C[a,b]$) as points. So, I think I should interpret this question as all possible metrics for any type of abstract set $X$, which only has two things--points--in it.

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Attempt:

No matter what the set $X$ considered is, as long as the metric $d$ defined on $X$ is maps $(x,y) \in X \times X$ to the nonnegative real numbers, (not including $+ \infty$), maps zero to zero, and is a non-affine function of $(x-y)$ it will suffice as a metric. ...

...all of these assumptions I'm making will just build to the definition of a metric it seems like.

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Maybe the author means a traditional set of points, i.e., finite tuples? I think I'm missing the spirit of the question.

Answer 1

You've given a description of a metric (though it's not quite correct), but you haven't described specifically what a metric on a space consisting of two points could be.

Suppose $X=\{a,b\}$. As you stated, the metric must give a value to $d(x,y)$ for every $(x,y) \in X \times X$. In other words, you need to specify $d(a,a), d(a,b),d(b,a)$, and $d(b,b)$. How much choice do you have for these values?

Your description of a metric is not quite correct for a few reasons. You say it must "map zero to zero," but the metric is a function of points $(x,y) \in X \times X$, so what you really mean is it maps points of the form $(x,x)$ to zero. You also say it must be a non-affine function of $(x-y)$, but it is clearer and more correct to say that if $x \neq y$ then $d(x,y)\neq 0$. Finally, your description is not sufficient to define a metric, because a metric must also be symmetric: $d(x,y)=d(y,x)$ for all $x,y \in X$.

Answer 2

If $X$ has only two points, then the triangle inequality property is a consequence of ($M1$) to ($M3$). Thus, any functions satisfy ($M1$) to ($M3$) is a metric on $X$. If $X$ has only one point, say, $x_0$, then the symmetry and triangle inequality property are both trivial. However, since we require $d(x_0, x_0) = 0$, any nonnegative function $f(x, y)$ such that $f(x_0, x_0) = 0$ is a metric on $X$.