Almost Injective Mappings of Totally Bounded Metric Spaces into Finite Dimensional Euclidean Spaces

Abstract

Let χ= be a metric space and let ε be a positive real number. Then a function f : X →Y is defined to be an ε -map if and only if for all y ∈ Y, the diameter of f -1 (y) is at most ε . In Theorem 10 we will give a new proof for the following well known fact: if χ is totally bounded, then for all ε there exists a finite number n and a continuous ε -map f ε : X→R n (here R n is the usual n -dimensional Euclidean space endowed with the Euclidean metric). If ε is “small”, then f ε is “almost injective”; and still exists even if χ has infinite covering dimension (in this case, n depends on ε , of course). Contrary to the known proofs, our proof technique is effective in the sense, that it allows establishing estimations for n in terms of ε and structural properties of χ .