We investigate some basic descriptive set theory for countably based completely quasi-metrizable topological spaces, which we refer to as quasi-Polish spaces. These spaces naturally generalize much of the classical descriptive set theory of Polish spaces to the non-Hausdorff setting. We show that a subspace of a quasi-Polish space is quasi-Polish if and only if it is Π2 in the Borel hierarchy. ...

By de Vries duality, the category of compact Hausdorff spaces is dually equivalent to the category of de Vries algebras (complete Boolean algebras endowed with a proximity-like relation). We provide an alternative “modal-like” duality by introducing the concept of a Gleason space, which is a pair (X,R), where X is an extremally disconnected compact Hausdorff space and R is an irreducible equiva...

Let  $varpi$ be a representation of the homogeneous space $G/H$, where $G$ be a locally compact group and  $H$ be a compact subgroup of $G$. For  an admissible wavelet $zeta$ for $varpi$  and $psi in L^p(G/H), 1leq p <infty$, we determine a class of bounded  compact operators  which are related to continuous wavelet transforms on homogeneous spaces and they are called localization operators.

In this article we introduce the concept of $z^circ$-filter on a topological space $X$. We study and investigate the behavior of $z^circ$-filters and compare them  with corresponding ideals, namely, $z^circ$-ideals of $C(X)$,  the ring of real-valued continuous functions on a completely regular Hausdorff space $X$. It is observed that $X$ is a compact space if and only if every $z^circ$-filter ...

For a topological space X, let L(X) be the modal logic of X where is interpreted as interior (and hence ♦ as closure) in X. It was shown in [6] that the modal logics S4, S4.1, S4.2, S4.1.2, S4.Grz, S4.Grzn (n ≥ 1), and their intersections arise as L(X) for some Stone space X. We give an example of a scattered Stone space whose logic is not such an intersection. This gives an affirmative answer ...

The contravariant powerset, and its generalisations Σ to the lattices of open subsets of a locally compact topological space and of recursively enumerable subsets of numbers, satisfy the Euclidean principle that φ ∧ F (φ) = φ ∧ F (>). Conversely, when the adjunction Σ(−) a Σ(−) is monadic, this equation implies that Σ classifies some class of monos, and the Frobenius law ∃x.(φ(x) ∧ ψ) = (∃x.φ(x...

We deal with two classes of locally compact sober spaces, namely, the class of locally spectral coherent spaces and the class of spaces in which every point has a closed spectral neighborhood (CSN-spaces, for short). We prove that locally spectral coherent spaces are precisely the coherent sober spaces with a basis of compact open sets. We also prove that CSN-spaces are exactly the locally spec...

In this paper we have characterized the epimorphisms in the full subcategory of Hausdorff fuzzy topological spaces (introduced by Srivastava et al.[10]) of the category FTS of fuzzy topological spaces and fuzzy continuous functions using the Salbanytype closure operator.

We introduce a classification of locally compact Hausdorff topological spaces with respect to the behavior $$ \sigma -compact subsets and, relying on this classification, we study properties corresponding $$C^*$$ -algebras in terms frame theory and {\mathscr A} operators Hilbert -modules; some pathological examples are constructed.