The based Kuratowski distance map into bounded continuous functions

Let $(M,d)$ be a nonempty metric space (Metric space: $d(x,y)=0$ iff $x=y$, symmetry, and the triangle inequality; pseudometric and ultrametric) and fix a basepoint $o\in M$. Inside the real function space ${\mathbb{R}}^M$ of The vector space $F^X$ of all functions $X\to F$ with pointwise operations, and $F^n$ as the case $X=n=\{0,1,\dots ,n-1\}$, write $C_b(M)$ for the set of bounded continuous real-valued functions on $M$, equipped with the supremum norm

$\mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\infty }}:={\sup }_{z\in M}\mathrm{\mid }f(z)\mathrm{\mid }\mathrm{.}$

For each $x\in M$, define a function $K_o(x):M\to \mathbb{R}$ by

$K_o(x)(z):=d(x,z)-d(o,z)(z\in M)\mathrm{.}$

The triangle inequality gives

$\mathrm{\mid }{K}_o(x)(z)\mathrm{\mid }=\mathrm{\mid }d(x,z)-d(o,z)\mathrm{\mid }\le d(x,o),$

so $K_o(x)$ is bounded. The same inequality in the $z$ variable shows that $K_o(x)$ is $2$-Lipschitz, hence continuous. Therefore $K_o(x)\in C_b(M)$ for every $x\in M$.

The resulting map

$K_o:M\to C_b(M),x\mapsto K_o(x),$

is the based Kuratowski distance map.

• The subtraction by $d(o,\cdot )$ is what keeps the target bounded when the metric space itself is unbounded.