I'd be surprised if there were any applications, for the simple reason that your definition of inverse limit for metric geometry doesn't translate well into topology. It's easy to disconnect points in the limit which aren't disconnected in the terms.

For example, fix a metric space (X, d) and define (X_n, d_n) = (X, 2ⁿd), with φ_{m, n} the identity on the underlying set X. Then the inverse limit would have underlying set X (essentially), but d_∞(x, y) = ∞ unless x = y.

Come to think of it, this seems to provide a counterexample to your statement, since the underlying space could easily be a polyhedron (with uncountable underlying set), while any discrete set which embeds into R^d is at most countable. But perhaps I've misunderstood.

I'd be surprised if there were any applications, for the simple reason that your definition of inverse limit for metric geometry doesn't translate well into topology. It's easy to disconnect points in the limit which aren't disconnected in the terms.

For example, fix a metric space (X, d) and define (X_n, d_n) = (X, 2ⁿd), with φ_{m, n} the identity on the underlying set X. Then the inverse limit would have underlying set X (essentially), but d_∞(x, y) = ∞ unless x = y.

Come to think of it, this seems to provide a counterexample to your statement, since the underlying space could easily be a polyhedron (with uncountable underlying set), while any discrete set which embeds into R^d is at most countable.

(Never mind, I see in your paper that you assume the space to be embedded is compact.)