### A topological lattice on the set of multifunctions.

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We prove that the interval topology of an Archimedean atomic lattice effect algebra $E$ is Hausdorff whenever the set of all atoms of $E$ is almost orthogonal. In such a case $E$ is order continuous. If moreover $E$ is complete then order convergence of nets of elements of $E$ is topological and hence it coincides with convergence in the order topology and this topology is compact Hausdorff compatible with a uniformity induced by a separating function family on $E$ corresponding to compact and cocompact elements....

We prove the following theorem: Given a⊆ω and $1\le \alpha <{\omega}_{1}^{CK}$, if for some $\eta <{\aleph}_{1}$ and all u ∈ WO of length η, a is ${\Sigma}_{\alpha}^{0}\left(u\right)$, then a is ${\Sigma}_{\alpha}^{0}$.We use this result to give a new, forcing-free, proof of Leo Harrington’s theorem: ${\Sigma}_{1}^{1}$-Turing-determinacy implies the existence of ${0}^{}$.

Let $A$ be a uniformly closed and locally m-convex $\Phi $-algebra. We obtain internal conditions on $A$ stated in terms of its closed ideals for $A$ to be isomorphic and homeomorphic to ${C}_{k}\left(X\right)$, the $\Phi $-algebra of all the real continuous functions on a normal topological space $X$ endowed with the compact convergence topology.

It is shown that for a metric space (M,d) the following are equivalent: (i) M is a complete ℝ-tree; (ii) M is hyperconvex and has unique metric segments.

Let $\tau $ be an uncountable regular cardinal and $G$ a ${T}_{1}$ topological group. We prove the following statements: (1) If $\tau $ is homeomorphic to a closed subspace of $G$, $G$ is Abelian, and the order of every non-neutral element of $G$ is greater than $5$ then $\tau \times \tau $ embeds in $G$ as a closed subspace. (2) If $G$ is Abelian, algebraically generated by $\tau \subset G$, and the order of every element does not exceed $3$ then $\tau \times \tau $ is not embeddable in $G$. (3) There exists an Abelian topological group $H$ such that ${\omega}_{1}$ is homeomorphic to a closed subspace...

In this paper we introduce and study new concepts of convergence and adherent points for fuzzy filters and fuzzy nets in the light of the $Q$-relation and the $Q$-neighborhood of fuzzy points due to Pu and Liu [28]. As applications of these concepts we give several new characterizations of the closure of fuzzy sets, fuzzy Hausdorff spaces, fuzzy continuous mappings and strong $Q$-compactness. We show that there is a relation between the convergence of fuzzy filters and the convergence of fuzzy nets similar...