Sági, G. and Nyiri, D. (2016) On Embeddings of Finite Metric Spaces. The Proceedings of the 13th International Scientific Conference on Informatics , Poprad, 18-20 November 2015, 227-231.
https://doi.org/10.1109/Informatics.2015.7377837

has been cited by the following article:

• 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.