Note on dimension theory for metric spaces

Asymptotic cones and Assouad-Nagata dimension

“…A metric space (X, ρ) is of Assouad-Nagata dimension (see [19] and [1]) at most n (notation: dim AN (X) ≤ n) if there is a constant c > 0 such that for any r > 0, there is a cover U r of X whose elements have diameter at most c · r and every r-ball B(x, r) intersects at most n + 1 elements of U r . Equivalently (see [7]), there is a constant K > 0 such that for any r > 0, X can be expressed as X = X 0 ∪ .…”

A study of metric-dependent dimension functions

“…Thus it is natural to seek an additional condition upon %{ with which the existence of the sequence does characterize dimension. Dowker-Hurewicz [2], Nagata [17] and Nagami [14] considered such a condition. This type of characterization theorem is one of the main foundations on which modern dimension theory has been built up.…”

Dimension and Metrization

“…As a matter of fact, we applied this metric to characterizing dimension of metric spaces in another way [7], [8]. That characterization theorem was simplified in separable cases by J. de Groot [2] as follows.…”