Related papers: Real compactness via real maximal ideals of $B_1(X…

In this paper, it is shown that a topological space $X$ is compact iff every maximal ideal of the power set ring $\mathcal{P}(X)$ converges to exactly one point of $X$. Then as an application, simple and ring-theoretic proofs are provided…

For a Tychonoff space $X$, $B_1(X)$ denotes the space of all Baire-one functions on $X$ endowed with the pointwise topology. We prove that the following assertions are equivalent: (1) $B_1(X)$ is a (semi-)Montel space, (2) $B_1(X)$ is a…

This paper studies the combinatorics of ideals which recently appeared in ergodicity results for analytic equivalence relations. The ideals have the following topological representation. There is a separable metrizable space $X$, a…

This paper focuses mainly on the ring of all bounded Baire one functions on a topological space. The uniform norm topology arises from the $\sup$-norm defined on the collection $B_1^*(X)$ of all bounded Baire one functions. With respect to…

If $I$ is an ideal in the ring $C(X)$ of all real valued continuous functions defined over a Tychonoff space $X$, then $X$ is called $I$-$pseudocompact$ if the set $X\setminus \bigcap Z[I]$ is a bounded subset of $X$. Corresponding to $I$,…

The set of all maximal ideals of the ring $\mathcal{M}(X,\mathcal{A})$ of real valued measurable functions on a measurable space $(X,\mathcal{A})$ equipped with the hull-kernel topology is shown to be homeomorphic to the set $\hat{X}$ of…

In this article, we continue our study of Baire one functions on a topological space $X$, denoted by $B_1(X)$ and extend the well known M. H. Stones's theorem from $C(X)$ to $B_1(X)$. Introducing the structure space of $B_1(X)$, it is…

One of the main obstacle to study compactness in topological spaces via ideals was the definition of ideal convergence of subsequences as in the existing literature according to which subsequence of an ideal convergent sequence may fail to…

Let $X$ be a completely regular topological space. We study closed ideals $H$ of $C_B(X)$, the normed algebra of bounded continuous scalar-valued mappings on $X$ equipped with pointwise addition and multiplication and the supremum norm,…

In this article, we continue our study of the ring of Baire one functions on a topological space $(X,\tau)$, denoted by $B_1(X)$ and extend the well known M. H. Stones's theorem from $C(X)$ to $B_1(X)$. Introducing the structure space of…

Let $X$ be a completely regular topological space. We assign to each (set theoretic) ideal of $X$ an (algebraic) ideal of $C_B(X)$, the normed algebra of continuous bounded complex valued mappings on $X$ equipped with the supremum norm. We…

We say that a Tychonoff space $X$ is a $\kappa$-space if it is homeomorphic to a closed subspace of $C_p(Y)$ for some locally compact space $Y$. The class of $\kappa$-spaces is strictly between the class of Dieudonn\'{e} complete spaces and…

A Tychonoff space $X$ is called $\kappa$-pseudocompact if for every continuous mapping $f$ of $X$ into $\mathbb{R}^\kappa$ the image $f(X)$ is compact. This notion generalizes pseudocompactness and gives a stratification of spaces lying…

This paper explores the duality between ideals of the ring $B_1(X)$ of all real valued Baire one functions on a topological space $X$ and typical families of zero sets, called $Z_B$-filters, on $X$. As a natural outcome of this study, it is…

The path component space of a topological space $X$ is the quotient space $\pi_0(X)$ whose points are the path components of $X$. We show that every Tychonoff space $X$ is the path-component space of a Tychonoff space $Y$ of weight…

Let $\mathcal C$ be a class of topological semigroups. A semigroup $X$ is called (1) $\mathcal C$-$closed$ if $X$ is closed in every topological semigroup $Y\in\mathcal C$ containing $X$ as a discrete subsemigroup, (2) $ideally$ $\mathcal…

As it was introduced by Tkachuk and Wilson, a topological space $X$ is cellular-compact if given any cellular, i.e. disjoint, family $\mathcal U$ of non-empty open subsets of $X$ there is a compact subspace $K\subset X$ such that $K\cap…

This article is a fundamental study in computable analysis. In the framework of Type-2 effectivity, TTE, we investigate computability aspects on finite and infinite products of effective topological spaces. For obtaining uniform results we…

The deck, $\mathcal{D}(X)$, of a topological space $X$ is the set $\mathcal{D}(X)=\{[X \setminus \{x\}]\colon x \in X\}$, where $[Y]$ denotes the homeomorphism class of $Y$. A space $X$ is (topologically) reconstructible if whenever…

A topological space $X$ is called $\Cal A$-real compact, if every algebra homomorphism from $\Cal A$ to the reals is an evaluation at some point of $X$, where $\Cal A$ is an algebra of continuous functions. Our main interest lies on…