It is shown that for every n ≥ 2 there exists an n-dimensional locally connected Polish space with Dimensionsgrad 1.

Let $X$ be a Hausdorff space and let $\mathscr{H}$ be one of the hyperspaces $CL\left(X\right)$, $𝒦\left(X\right)$, $ℱ\left(X\right)$ or ${ℱ}_n\left(X\right)$ ($n$ a positive integer) with the Vietoris topology. We study the following disconnectedness properties for $\mathscr{H}$: extremal disconnectedness, being a $F^{\text{'}}$-space, $P$-space or weak $P$-space and hereditary disconnectedness. Our main result states: if $X$ is Hausdorff and $F\subset X$ is a closed subset such that (a) both $F$ and $X-F$ are totally disconnected, (b) the quotient $X/F$ is hereditarily disconnected, then $𝒦\left(X\right)$ is hereditarily disconnected. We also...

Utilizing the discrete homotopy methods developed for uniform spaces by Berestovskii-Plaut, we define the critical spectrum Cr(X) of a metric space, generalizing to the non-geodesic case the covering spectrum defined by Sormani-Wei and the homotopy critical spectrum defined by Plaut-Wilkins. If X is geodesic, Cr(X) is the same as the homotopy critical spectrum, which differs from the covering spectrum by a factor of 3/2. The latter two spectra are known to be discrete for compact geodesic spaces,...