Given a metric space $(X,d)$ and a subset $S\subset X$, the projection $P_S$ onto $S$ is well-defined as a set valued function. I am interested in learning more about properties of these projections such as uniqueness, continuity, proximinality, etc.
Unfortunately, almost every paper I can find on this topic considers the special case where $X$ is a Banach space. I am interested specifically in the case where $X$ is a general metric space. Any pointers to papers or monographs would be appreciated.
Note. Apparently Ivan Singer has some papers from the 1960s on this, but I can’t track down the full text of the papers online. For example:
 Some remarks on approximative compactness. Rev. Roum. Math. Pures Appl. 9(1964), 167-177
References on metric spaces. Here are a couple papers I have found specifically regarding metric spaces:
 Khalil, R. (1988). Best approximation in metric spaces. Proceedings of the American mathematical society, 103(2), 579-586.
 Gupta, S., & Narang, T. D. (2017). STRONG PROXIMINALITY IN METRIC SPACES. Novi Sad J. Math, 47(2), 107-116.
References on Banach spaces. Just in case, here are useful some references on the Banach space version.
 D. D. Repovš and P.V. Semenov, Continuous Selections of Multivalued Mappings, Kluwer Academic Publishers, Dordrecht 1998. http://www.pef.uni-lj.si/repovs/knjige/kazalo_book.htm See Chapter C.6 on metric projections
 Alber, Y. I. (1996). A bound for the modulus of continuity for metric projections in a uniformly convex and uniformly smooth Banach space. journal of approximation theory, 85(3), 237-249.
 Deutsch, F., & Lambert, J. M. (1980). On continuity of metric projections. Journal of Approximation Theory, 29(2), 116-131.