TITLE:
Almost Injective Mappings of Totally Bounded Metric Spaces into Finite Dimensional Euclidean Spaces
AUTHORS:
Gábor Sági
KEYWORDS:
Totally Bounded Metric Spaces, Dimension Theory, Finite Dimensional Euclidean Spaces, ε-Mapping
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.9 No.6,
June
30,
2019
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 fε: X→Rn (here Rn is the usual n-dimensional Euclidean space endowed
with the Euclidean metric). If ε is “small”, then fε 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.