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Let $\Gamma(X)$ be the inverse semigroup of partial homeomorphisms between open subsets of a compact metric space $X$. There is a topology, denoted $\tau_{hco}$, that makes $\Gamma(X)$ a topological inverse semigroup. We address the question of whether $\tau_{hco}$ is Polish. For a 0-dimensional compact metric space $X$, we prove that $(\Gamma(X), \tau_{hco})$ is Polish by showing that it is topologically isomorphic to a closed subsemigroup of the Polish symmetric inverse semigroup $I(\N)$. We present examples, similar to the classical Munn semigroups, of Polish inverse semigroups consisting of partial isomorphism on lattices of open sets.

We address some phenomena about the interaction between lower semicontinuous submeasures on $\mathbb{N}$ and $F_{\sigma}$ ideals. We analyze the pathology degree of a submeasure and present a method to construct pathological $F_\sigma$ ideals. We give a partial answers to the question of whether every nonpathological tall $F_\sigma$ ideal is Kat\v{e}tov above the random ideal or at least has a Borel selector. Finally, we show a representation of nonpathological $F_\sigma$ ideals using sequences in Banach spaces., Comment: arXiv admin note: substantial text overlap with arXiv:2111.10598

We study a property about Polish inverse semigroups similar to the classical theorem of Pettis about Polish groups. In contrast to what happens with Polish groups, not every Polish inverse semigroup have the Pettis property. We present several examples of Polish inverse subsemigroup of the symmetric inverse semigroup I(N) of all partial bijections between subsets of N. We also study whether our examples satisfy automatic continuity.

We present some generalizations of the well-known correspondence, found by R. Exel, between partial actions of a group $G$ on a set $X$ and semigroup homomorphism of $S(G)$ on the semigroup $I(X)$ of partial bijections of $X,$ being $S(G)$ an inverse monoid introduced by Exel. We show that any unital premorphism $\theta:G\to S$, where $S$ is an inverse monoid, can be extended to a semigroup homomorphism $\theta^*:T\to S$ for any inverse semigroup $T$ with $S(G)\subseteq T\subseteq P^*(G)\times G,$ being $P^*(G)$ the semigroup of non-empty subset of $G$, and such that $E(S)$ satisfies some lattice theoretical condition. We also consider a topological version of this result. We present a minimal Hausdorff inverse semigroup topology on $\Gamma(X)$, the inverse semigroup of partial homeomorphism between open subsets of a locally compact Hausdorff space $X$.

Let ${\bf x}=(x_n)_n$ be a sequence in a Banach space. A set $A\subseteq \mathbb{N}$ is perfectly bounded, if there is $M$ such that $\|\sum_{n\in F}x_n\|\leq M$ for every finite $F\subseteq A$. The collection $B({\bf x})$ of all perfectly bounded sets is an ideal of subsets of $\mathbb{N}$. We show that an ideal $\mathcal{I}$ is of the form $B({\bf x})$ iff there is a non pathological lower semicontinuous submeasure $\varphi$ on $\mathbb{N}$ such that $\mathcal{I} =FIN(\varphi)=\{A\subseteq \mathbb{N}: \;\varphi(A)<\infty\}$. We address the questions of when $FIN(\varphi)$ is a tall ideal and has a Borel selector. We show that in $c_0$ the ideal $B({\bf x})$ is tall iff $(x_n)_n$ is weakly null, in which case, it also has a Borel selector., Comment: Substitute by preprint arXiv 2211.01544

We study a reconstruction problem for colorings. Given a finite or countable set $X$, a coloring on $X$ is a function $\varphi: [X]^{2}\to \{0,1\}$, where $[X]^{2}$ is the collection of all 2-elements subsets of $X$. A set $H\subseteq X$ is homogeneous for $\varphi$ when $\varphi$ is constant on $[H]^2$. Let $hom(\varphi)$ be the collection of all homogeneous sets for $\varphi$. The coloring $1-\varphi$ is called the complement of $\varphi$. We say that $\varphi$ is {\em reconstructible} up to complementation from its homogeneous sets, if for any coloring $\psi$ on $X$ such that $hom(\varphi)=hom(\psi)$ we have that either $\psi=\varphi$ or $\psi=1-\varphi$. We present several conditions for reconstructibility and non reconstructibility. We show that there is a Borel way to reconstruct a coloring from its homogeneous sets.

Given a hereditarily meager ideal $\mathcal{I}$ on a countable set $X$ we use Martin's axiom for countable posets to produce a zero-dimensional maximal topology $\tau^\mathcal{I}$ on $X$ such that $\tau^\mathcal{I}\cap \mathcal{I}=\{\emptyset\}$ and, moreover, if $\mathcal{I}$ is $p^+$ then $\tau^\mathcal{I}$ is selectively separable (SS) and if $\mathcal{I}$ is $q^+$, so is $\tau^\mathcal{I}$. In particular, we obtain regular maximal spaces satisfying all boolean combinations of the properties SS and $q^+$., Comment: 17 pages

We study some variations of the product topology on families of clopen subsets of $2^{\mathbb{N}}\times\mathbb{N}$ in order to construct countable nodec regular spaces (i.e. in which every nowhere dense set is closed) with analytic topology which in addition are not selectively separable and do not satisfy the combinatorial principle $q^+$.

An ideal on a set $X$ is a collection of subsets of $X$ closed under the operations of taking finite unions and subsets of its elements. Ideals are a very useful notion in topology and set theory and have been studied for a long time. We present a survey of results about ideals on countable sets and include many open questions.

Given a dendrite $X$ and a continuous map $f\colon X\to X$, we show the following are equivalent: (i) $\omega_f$ is continuous and $\overline{\mathrm{Per}(f)}=\bigcap_{n\in\mathbb{N}}f^n(X)$; (ii) $\omega(x,f)=\Omega(x,f)$ for each $x\in X$; and (iii) $f$ is equicontinuous. Furthermore, we present some examples illustrating our results.

Let $X$ be a compact metric countable space, let $f:X\to X$ be a homeomorphism and let $E(X,f)$ be its Ellis semigroup. Among other results we show that the following statements are equivalent: (i) $(X,f)$ is equicontinuous, (ii) $(X,f)$ is distal and (iii) every point is periodic. We use this result to give a direct proof of a theorem of Ellis saying that $(X,f)$ is distal if, and only if, $E(X,f)$ is a group.

We present a method for completing a premetric space, in the sense introduced by F. Richman in the context of constructive Mathematics without countable choice.

Several tasks in artificial intelligence require to be able to find models about knowledge dynamics. They include belief revision, fusion and belief merging, and abduction. In this paper we exploit the algebraic framework of mathematical morphology in the context of propositional logic, and define operations such as dilation or erosion of a set of formulas. We derive concrete operators, based on a semantic approach, that have an intuitive interpretation and that are formally well behaved, to perform revision, fusion and abduction. Computation and tractability are addressed, and simple examples illustrate the typical results that can be obtained.

We show that the Nash-Williams theorem has a uniform version and that the Galvin theorem does not. We show that there is an $F_\sigma$ tall ideal on $\mathbb{N}$ without a Borel selector and also construct a $\mathbf\Pi^1_2$ tall ideal without a tall closed subset.

We show that the following properties are preserved under inverse limits: countable fan-tightness, q+, discrete generation and selective separability. We also present several examples based on inverse limits of countable spaces.

We show that the enveloping space $X_G$ of a partial action of a Polish group $G$ on a Polish space $X$ is a standard Borel space, that is to say, there is a topology $\tau$ on $X_G$ such that $(X_G, \tau)$ is Polish and the quotient Borel structure on $X_G$ is equal to $Borel(X_G,\tau)$. To prove this result we show a generalization of a theorem of Burgess about Borel selectors for the orbit equivalence relation induced by a group action and also show that some properties of the Vaught's transform are valid for partial actions of groups.

Let $X$ be a separable metrizable space. We establish a criteria for the existence of a metrizable globalization for a given continuous partial action of a separable metrizable group $G$ on $X.$ If $G$ and $X$ are Polish spaces, we show that the globalization is also a Polish space. We also show the existence of an universal globalization for partial actions of Polish groups.

Given a continuum $X$, for each $A\subseteq X$, the Jones' set function $\mathcal{T}$ is defined by $\mathcal{T}(A)=\{x\in X : \text{for each subcontinuum }K\text{ such that }x\in \textrm{Int}(K), \text{ then }K\cap A\neq\emptyset\}.$ We show that $\mathcal{D}=\{\mathcal{T}(\{x\}):x\in X\}$ is decomposition of $X$ when $\mathcal{T}$ is continuous. We present a characterization of the continuity of $\mathcal{T}$ and answer several open questions posed by D. Bellamy.

We study Borel ideals $I$ on $\mathbb{N}$ with the Fr\'echet property such its orthogonal $I^\perp$ is also Borel (where $A\in I^\perp$ iff $A\cap B$ is finite for all $B\in I$ and $I$ is Fr\'echet if $I=I^{\perp\perp}$). Let $\mathcal{B}$ be the smallest collection of ideals on ${\mathbb{N}}$ containing the ideal of finite sets and closed under countable direct sums and orthogonal. All ideals in $\mathcal{B}$ are Fr\'echet, Borel and have Borel orthogonal. We show that $\mathcal{B}$ has exactly $\aleph_1$ non isomorphic members. The family $\mathcal{B}$ can be characterized as the collection of all Borel ideals which are isomorphic to an ideal of the form $I_{wf}\up A$, where $I_{wf}$ is the ideal on $\mathbb{N}^{<\omega}$ generated by the wellfounded trees. Also, we show that $A\subseteq \mathbb{Q}$ is scattered iff $WO({\mathbb{Q}})\up A$ is isomorphic to an ideal in $\mathcal{B}$, where $WO(\mathbb{Q})$ is the ideal of well founded subset of $\mathbb{Q}$.

We present results about the Cantor-Bendixson index of some subspaces of a uniform family F of finite subsets of natural numbers with respect to the lexicographic order topology. As a corollary of our results we get that for any omega-uniform family F the restriction F|M is homeomorphic to F iff M contains intervals of arbitrary length of consecutive integers. We show the connection of these results with a topological partition problem of uniform families.

It is shown that Matet's characterization of the Ramsey property relative to a selective co-ideal $\mathcal{H}$, in terms of games of Kastanas, still holds if we consider semiselectivity instead of selectivity. Moreover, we prove that a co-ideal $\mathcal{H}$ is semiselective if and only if Matet's game-theoretic characterization of the $\mathcal{H}$-Ramsey property holds. This lifts Kastanas's characterization of the classical Ramsey property to its optimal setting, from the point of view of the local Ramsey theory and gives a game-theoretic counterpart to a theorem of Farah \cite{far}, asserting that a co-ideal $\mathcal{H}$ is semiselective if and only if the family of $\mathcal{H}$-Ramsey subsets of $\N^{[\infty]}$ coincides with the family of those sets having the abstract $\mathcal{H}$-Baire property. Finally, we show that under suitable assumptions, for every semiselective co-ideal $\mathcal H$ all sets of real numbers are $\mathcal H$-Ramsey., Comment: 11 pages