The problem, whether every topological space has a weak compact reflection, was answered by M. Hušek in the negative. Assuming normality, M. Hušek fully characterized the spaces having a weak reflection in compact spaces as the spaces with the finite Wallman remainder. In this paper we prove that the assumption of normality may be omitted. On the other hand, we show that some covering properties kill the weak reflectivity of a noncompact topological space in compact spaces.

Let $X$ be a finite graph. Let $C\left(X\right)$ be the hyperspace of all nonempty subcontinua of $X$ and let $\mu :C\left(X\right)\to ℝ$ be a Whitney map. We prove that there exist numbers $0<T_0<T_1<T_2<\cdots <T_M=\mu \left(X\right)$ such that if $T\in (T_{i-1},T_i)$, then the Whitney block ${\mu }^{-1}(T_{i-1},T_i)$ is homeomorphic to the product ${\mu }^{-1}\left(T\right)\times (T_{i-1},T_i)$. We also show that there exists only a finite number of topologically different Whitney levels for $C\left(X\right)$.

It is shown that there is no Whitney map on the hyperspace $2^X$ for non-metrizable Hausdorff compact spaces X. Examples are presented of non-metrizable continua X which admit and ones which do not admit a Whitney map for C(X).

We prove the following results. (i) Let X be a continuum such that X contains a dense arc component and let D be a dendrite with a closed set of branch points. If f:X → D is a Whitney preserving map, then f is a homeomorphism. (ii) For each dendrite D' with a dense set of branch points there exist a continuum X' containing a dense arc component and a Whitney preserving map f':X' → D' such that f' is not a homeomorphism.

Given a metric space ⟨X,ρ⟩, consider its hyperspace of closed sets CL(X) with the Wijsman topology ${\tau }_{W\left(\rho \right)}$. It is known that $⟨CL\left(X\right),{\tau }_{W\left(\rho \right)}⟩$ is metrizable if and only if X is separable, and it is an open question by Di Maio and Meccariello whether this is equivalent to $⟨CL\left(X\right),{\tau }_{W\left(\rho \right)}⟩$ being normal. We prove that if the weight of X is a regular uncountable cardinal and X is locally separable, then $⟨CL\left(X\right),{\tau }_{W\left(\rho \right)}⟩$ is not normal. We also solve some questions by Cao, Junnila and Moors regarding isolated points in Wijsman hyperspaces.