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

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→Rⁿ (here Rⁿ 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.