TITLE: What is a virtual point of a metric space?
QUESTION [21 upvotes]: Suppose $(X, d)$ is a metric space and $u \colon X → \mathbb{R}$. Then $u$ will be called a virtual point of $X$ if, and only if, $u$ satisfies the following three conditions:

$u(x) - u(y) \leq d(x,y) \leq u(x) + u(y)$,
$\inf_{x \in X} u(x) = 0$, and
$u(z) \neq 0$.


What is the intuition behind such a definition? This is from Mícheál Ó Searcóid’s book where they have proposed an equivalent condition of $X$ having no virtual points and $X$ being complete. Can someone please explain.

REPLY [14 votes]: Conditions 1 and 3 in the definition you quote are equivalent to saying that, if you added an extra point $a \notin X$ to $X$ and defined $$d(a,x) = d(x,a) = u(x)$$ for all $x \in X$, and $d(a,a) = 0$, then the set $X \cup \{a\}$, equipped with the expanded distance function $d$ thus defined, would still be a metric space.  Specifically, condition 1 ensures that the expanded metric $d$ still satisfies the triangle inequality, while condition 3 ensures that $d(a,x) ≠ 0$ for any pre-existing point $x \in X$.

As for condition 2, what it amounts to is requiring that the extra point $a$ thus added will not be an isolated point, but can be approached arbitrarily closely by existing points in $X$.  This additional condition is essential if your goal is to define the absence of "virtual points" as equivalent to completeness, since without it all metric spaces would have infinitely many virtual points.
As an example of how to construct an "isolated virtual point" violating condition 2, take any point $z \in X$ and any real number $c > 0$ and let $u(x) = d(x, z) + c$ for all $x \in X$.  Clearly any such function $u$ will satisfy conditions 1 and 3, but cannot satisfy condition 2.
(The "geometric intuition" behind this definition is that the new point $a$ is placed at distance $c$ from $z$, and its distance from other points in $X$ is defined by assuming that you can only reach $a$ from $X$ by going through $z$.  Obviously other arrangements are possible too: for example, we could make the new point $a$ reachable via $n$ points in $X$ instead of just one, i.e. pick $z_1, \dots, z_n \in X$ and $c_1, \dots, c_n \in \mathbb R^+$, making sure that $c_i + c_j ≥ d(z_i, z_j)$ for all $1 ≤ i, j ≤ n$ so that condition 1 will still be satisfied, and let $u(x) = \min_{1 ≤ i ≤ n}(d(x, z_i) + c_i)$.)