Representations of compact subsets of \(\mathbb{R}^n\) (Q5960418)

Throughout this review let \(X\) be a \(k\)-dimensional compact subset of \(\mathbb{R}^n\). It is well known that \(X\) can be embedded in a \(k\)-dimensional Menger subset \(M_k^n\). It is also known that \(X\) need not contain a subset homeomorphic to the \(k\)-dimensional cube \(I^k\). When this is the case we say that \(X\) has the property \(EI(k)\). The authors show the following criterion for \(X\) to have property \(EI(k)\) as follows: NEWLINENEWLINENEWLINETheorem 9. There exists a dense \(F_\sigma\) subset \(C_k^n\) of \(M_k^n\) with the following property. If there exists an embedding \(f:X \to M_k^n\) such that \(\dim f(X)\cap C_k^n = k\), then \(X\) has the property \(EI(k)\). NEWLINENEWLINENEWLINEMoreover, when \(k\) is large enough (i.e. \(n < 2k\)) they show the following interesting result by using a classical result due to K. A. Sitnikov: NEWLINENEWLINENEWLINETheorem 10. When \(n < 2k\) and \(X\) has \(EI(k)\), then every embedding \(f:X \to M_k^n\) satisfies the condition of Theorem 9 (i.e. \(\dim f(X)\cap C_k^n = k\)).