Questions on weakly infinite-dimensional spaces

Publisher Summary This chapter discusses questions on weakly infinite-dimensional spaces. Weak infinite dimensionality was introduced by Alexandroff in 1948. The first results in this area were obtained by Sklyarenko and Levshenko in 1959. A great contribution to the theory of weakly infinite-dimensional spaces was made in 1981 by R. Pol, who constructed an example of a compact metrizable weakly infinite-dimensional space that is not countable-dimensional. In 1974, Haver introduced the C property for metric spaces and proved that every locally contractible metric space that is a union of countably many compact sets with property C is an ANR space. In 1978, Addis and Gresham gave a topological definition of C -spaces. The C -spaces proved to play an important role in topology. In particular, Ancel showed that any cell-like map from a compact metrizable space onto a C- space is a hereditary shape equivalence. Consequently, every infinite‑dimensional compact C-space has infinite cohomological dimension c -dim ℤ . One of the most important problems concerning infinite-dimensional spaces was whether any weakly infinite-dimensional compact space is a C -space. The questions considered in the chapter are related to new classes of spaces, which are intermediate between the classes of weakly infinite-dimensional spaces and C -spaces.