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Abstract

For each natural number $n \geq 1$ and each pair of ordinals $\alpha,\beta$ with $n \leq \alpha \leq \beta \leq \omega({\mathfrak c}^+)$, where $\omega({\mathfrak c}^+)$ is the first ordinal of cardinality ${\mathfrak c}^+$, we construct a continuum $S_{n,\alpha,\beta}$ such that

(a) $\dim S_{n,\alpha,\beta}=n$;

(b) $\mathop{{\rm trDg}}\nolimits S_{n,\alpha,\beta}=\mathop{{\rm trDgo}}\nolimits S_{n,\alpha,\beta}=\alpha$;

(c) $\mathop{\rm trind} S_{n,\alpha,\beta}=\mathop{{\rm trInd}_0}\nolimits S_{n,\alpha,\beta}=\beta$;

(d) if $\beta<\omega({\mathfrak c}^+)$, then $S_{n,\alpha,\beta}$ is separable and first countable;

(e) if $n=1$, then $S_{n,\alpha,\beta}$ can be made chainable or hereditarily decomposable;

(f) if $\alpha = \beta<\omega({\mathfrak c}^+)$, then $S_{n,\alpha,\beta}$ can be made hereditarily indecomposable;

(g) if $n=1$ and $\alpha = \beta<\omega({\mathfrak c}^+)$, then $S_{n,\alpha,\beta}$ can be made chainable and hereditarily indecomposable.

In particular, we answer the question raised by Chatyrko and Fedorchuk whether every non-degenerate chainable space has Dimensionsgrad equal to $1$. Moreover, we establish results that enable us to compute the Dimensionsgrad of a number of spaces constructed by Charalambous, Chatyrko, and Fedorchuk.