Previously asked and bountied at MSE:
For complete metric spaces $X,Y$, write $X\trianglelefteq Y$ iff for every complete metric space $Z$ such that the Gromov-Hausdorff distance between $X$ and $Z$ is achieved by some embedding, the Gromov-Hausdorff distance between $Y$ and $Z$ is also achieved by some embedding. For example, this answer by James Hanson shows that the one-point space is strictly $\triangleright$ a two-point space.
I'm mostly interested in the interaction between $\trianglelefteq$ and compactness. Specifically, suppose $X\trianglelefteq Y$ and $X$ is compact. Must $Y$ be compact as well? Such a $Y$ must achieve its Gromov-Hausdorff distance with every compact space, and this seems like a fairly strong requirement; however, I can't seem to conclude compactness from this alone (or even the stronger hypothesis actually being asked about here).
There are several additional questions I'm interested in about $\trianglelefteq$ (e.g. what's the ordertype of $\trianglelefteq$ on the compact spaces?), but this seems like a good place to start.
