Metric constructions of topological invariants

We present a general mechanism for obtaining topological invariants from metric constructs. In more detail, we describe a process which, under very mild conditions, produces topological invariants out of a construction on a metric space together with a choice of scale (a non-negative value at each point of the space). Through Flagg's metric formalism of topology the results are valid for all topological spaces, not just the metrizable ones. We phrase the result in much greater generality than required for the topological applications, using the language of fibrations. We show that ordinary topological connectedness arises metrically, and we obtain metrically defined theories of homology and of homotopy.