Axiomatic method of partitions in the theory of Nöbeling spaces. III. Consistency of the axiom system

A short proof of Toruńczyk’s characterization theorems

“…These results, first proved by Toruńczyk [13,14,15], are widely known and were applied in diverse settings. They also inspired the characterization results of the universal Menger spaces by Bestvina [4] and the recent work on Nöbeling spaces [1,2,3], [12], [8,9].…”

A construction of {N}

“…By recursive application of Lemma 5.2 we construct a sequence of maps g i : X → Y satisfying the following conditions. (2) and by Lemma 4.1, the sequence g i is uniformly convergent. As inverse limit of complete spaces, X is complete.…”

“…In the 1980s it was conjectured that ν n is an n-dimensional analog of the Hilbert space 2 . The conjecture was inspired by the observation that ν n satisfies n-dimensional analogs of properties that topologically characterize the Hilbert space.…”

Local $k$-connectedness of an inverse limit of polyhedra

“…(1) If f is surjective, then the covers f −1 (G) and G are isomorphic. (2) If G is open / closed / locally finite / locally finite-dimensional, then so is f −1 (G).…”

“…The Nöbeling space characterization theorem [3,1,2,10,12] states that if a space is strongly universal in the class of n-dimensional separable complete metric spaces and is k-connected and locally k-connected for each k < n, then it is homeomorphic to the n-dimensional Nöbeling space N 2n+1 n . In geometric group theory many spaces arise naturally as inverse limits of polyhedra (simplicial complexes endowed with the metric topology).…”