There exists a result previously stated that shows that a non-negative real valued function $\hat{d}:X\times X\rightarrow \mathbb{R}$ that satisfies symmetry and $d(x,x)=0$ (that is, different elements from the space are allowed to have distance zero) can be extended to a pseudometric? There is a result that a on the same conditions above, plus $d(x,y)=0\Rightarrow x=y$, then $\hat{d}$ can be extended to a metric.

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Usually under an extension of a function $f$ defined on a set $X$ (or a pseudometric $d$ defined on $X\times X$) understood a function $\bar f$ defined on a set $Y\subset X$ (resp. a pseudometric $\bar d$ defined on $Y\times Y$), such that a restriction $\bar f|X$ coincides with $f$ (resp. $\bar d|X\times X=f$).

I cite the beginning of my student paper “On Extension of (Pseudo-)Metrics from Subgroup of Topological Group onto the Group”

“The problem of extensions of functions from subobjects to objects in various categories was considered by many authors. The classic Tietze-Urysohn theorem on extensions of functions from a closed subspace of a topological space and its generalizations belong to the known results. Hausdorff [Hau] showed that every metric from a closed subspace of a metrizable space can be extended onto the space. Isbell [Isb] showed that every bounded uniform metric from a subspace of a uniform space can be extended onto the space. The linear operators extending metrics from a closed subspace of a metrizable space onto the space were considered in, e.g., [Bes, Zar]".

If we have a symmetric non-negative function $d$ on $X\times X$ such that $d(x,x)=0$ for each $x\in X$, a standard way to modify $d$ to a pseudometric $d’\le d$ is to put

$$d’(x,y)=\inf\left\{\sum_{i=1}^{n} d(x_{i-1},x_i):x_1,\dots, x_n\in X, x_0=x, x_n=y\right\}.$$

Remark, that $d’$ may fail to be a metric even when $d(x,y)=0\Rightarrow x=y$ for each $x,y\in X$.

References

[Bes] Bessaga C., Functional analytic aspects of geometry. Linear extending of metrics and related problems, in: Progress of Functional Analysis, Proc. Peniscola Meeting 1990 on the 60th birthday of Professor M. Valdivia, North-Holland, Amsterdam (1992) 247-257.

[Hau] Hausdorff F., Erweiterung einer Homömorpie, - Fund. Math., 16 (1930), 353--360.

[Isb] Isbell J.R. On finite-dimensional uniform spaces, - Pacific J. of Math., 9 (1959), 107-121.

[Zar] Zarichnyi M., Regular Linear Operators Extending Metrics: a Short Proof, Bull. Pol. Ac.:Math., 44, (1996), 267--269.

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