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Let S(X) denote the set of all closed subsets of a topological space X, and C(X) the set of all continuous mappings f:X → X. A family 𝓐 ⊆ S(X) is called reflexive if there exists ℱ ⊆ C(X) such that 𝓐 = {A ∈ S(X): f(A) ⊆ A for every f ∈ ℱ}. We investigate conditions ensuring that a family of closed subsets is reflexive.

Let $Con{v}_F\left(ℝⁿ\right)$ be the space of all non-empty closed convex sets in Euclidean space ℝ ⁿ endowed with the Fell topology. We prove that $Con{v}_F\left(ℝⁿ\right)\approx ℝⁿ\times Q$ for every n > 1 whereas $Con{v}_F\left(ℝ\right)\approx ℝ\times$.

For a topological space $X$, let $S\left(X\right)$ denote the set of all closed subsets in $X$, and let $C\left(X\right)$ denote the set of all continuous maps $f:X\to X$. A family $𝒜\subseteq S\left(X\right)$ is called reflexive if there exists $𝒞\subseteq C\left(X\right)$ such that $𝒜=\{A\in S(X):f(A)\subseteq A$ for every $f\in 𝒞\}$. Every reflexive family of closed sets in space $X$ forms a sub complete lattice of the lattice of all closed sets in $X$. In this paper, we continue to study the reflexive families of closed sets in various types of topological spaces. More necessary and sufficient conditions for certain families of closed...

A topological space $X$ is called mesocompact (sequentially mesocompact) if for every open cover $𝒰$ of $X$, there exists an open refinement $𝒱$ of $𝒰$ such that $\{V\in 𝒱:V\cap K\ne \varnothing \}$ is finite for every compact set (converging sequence including its limit point) $K$ in $X$. In this paper, we give some characterizations of mesocompact (sequentially mesocompact) spaces using selection theory.