For a multivalued map $\varphi :Y⊸(X,\tau )$ between topological spaces, the upper semifinite topology $𝒜\left(\tau \right)$ on the power set $𝒜\left(X\right)=\{A\subset X:A\ne \varnothing \}$ is such that $\varphi$ is upper semicontinuous if and only if it is continuous when viewed as a singlevalued map $\varphi :Y\to \left(𝒜\right(X),𝒜(\tau \left)\right)$. In this paper, we seek a result like this from a reverse viewpoint, namely, given a set $X$ and a topology $\Gamma$ on $𝒜\left(X\right)$, we consider a natural topology $ℛ\left(\Gamma \right)$ on $X$, constructed from $\Gamma$ satisfying $ℛ\left(\Gamma \right)=\tau$ if $\Gamma =𝒜\left(\tau \right)$, and we give necessary and sufficient conditions to the upper semicontinuity of a multivalued...

For a space $Z$, we denote by $ℱ\left(Z\right)$, $𝒦\left(Z\right)$ and ${ℱ}_2\left(Z\right)$ the hyperspaces of non-empty closed, compact, and subsets of cardinality $\le 2$ of $Z$, respectively, with their Vietoris topology. For spaces $X$ and $E$, $C_p(X,E)$ is the space of continuous functions from $X$ to $E$ with its pointwise convergence topology. We analyze in this article when $ℱ\left(Z\right)$, $𝒦\left(Z\right)$ and ${ℱ}_2\left(Z\right)$ have continuous selections for a space $Z$ of the form $C_p(X,E)$, where $X$ is zero-dimensional and $E$ is a strongly zero-dimensional metrizable space. We prove that $C_p(X,E)$ is weakly orderable...

We consider when one-to-one continuous mappings can improve normality-type and compactness-type properties of topological spaces. In particular, for any Tychonoff non-pseudocompact space $X$ there is a $\mu$ such that $X^{\mu }$ can be condensed onto a normal ($\sigma$-compact) space if and only if there is no measurable cardinal. For any Tychonoff space $X$ and any cardinal $\nu$ there is a Tychonoff space $M$ which preserves many properties of $X$ and such that any one-to-one continuous image of $M^{\mu }$, $\mu \le \nu$, contains a...