Papers

This article investigates the coprimality topology on the set of non-zero Gaussian integers, $\mathbb{Z}[i]\setminus\{0\}$, generated by the arithmetic basis $\sigma_\alpha=\{\beta\neq 0 : \gcd(\alpha,\beta)\sim 1\}$. By analyzing prime supports up to associates, we establish that two points are topologically indistinguishable if and only if their supports coincide. We demonstrate that the space is both hyperconnected and ultraconnected, and we explicitly characterize its Kolmogorov quotient $X$ as the space of finite subsets of associate classes of Gaussian primes, endowed with the basis $\mathcal{O}_F=\{S\in X:S\cap F=\emptyset\}$ for $F\in X$. Finally, we demonstrate that the set of Gaussian primes is dense in $\mathbb{Z}[i]\setminus\{0\}$ with respect to the coprimality topology, thereby establishing a topological proof of the infinitude of Gaussian primes.

Following a suggestion in a paper by Erné, Picado and Pultr, we establish the category of basic zero-dimensional spaces and their basic continuous maps. The objects are closure spaces with a distributive closure system containing a specified meet-base of complemented members that make the category a common generalization of those of closure spaces, zero-dimensional topological spaces, Heyting semilattices, and locales (frames). We treat images, preimages, continuity, openness and closedness of maps between basic zero-dimensional spaces. Among our main results, we have a useful description of preimages for basic continuous maps (analogous to the one for localic preimage maps as coframe homomorphisms), and a Joyal-Tierney type theorem for basic open maps. A novel aspect of the category of basic zero-dimensional spaces is a duality principle that, among other things, enables results for basic closed maps to be obtained for free from those for basic open maps.

Let $X$ be a topological space. For each transfinite cardinal number $\aleph_\alpha$, we define a topology $C_{\aleph_\alpha}(X)$ on the ring $C(X)$. With $\aleph_\alpha=\aleph_0$, $C_{\aleph_\alpha}(X)$ reduces to the space $C_p(X)$. We prove that for $\aleph_\alpha\geq \aleph_1$, $C_{\aleph_\alpha}(X)$ is pathwise connected if and only if it is connected if and only if $X$ is pseudocompact. Here, we define $\aleph_\alpha$-separable space and furthermore, we show that $X$ is $\aleph_\alpha$-separable when and only when $C_{\aleph_\alpha}(X)$ is metrizable when and only when it is a sequential space. Later we introduce two new cardinal functions, namely $cc_{\aleph_\alpha}(X)$ and $ac_{\aleph_\alpha}(X)$ which turned out to be the character and pseudocharacter of $C_{\aleph_\alpha}(X)$ respectively. At the end of this article we show that a number cardinal functions associated with the space $C_{\aleph_\alpha}(X)$ are equal.

Deep Operator Networks (DeepONets; arXiv:1910.03193) typically encode an input function through point values on a fixed discretization. Building on the Topological DeepONet framework of Ismailov (arXiv:2603.11972), we replace point samples by continuous linear functionals drawn from the continuous dual of a Hausdorff locally convex space $({V},\{p_\alpha\}_{\alpha\in A})$, whose topology is generated by a point-separating family of seminorms rather than a single norm, and develop fixed and adaptive functional measurement systems. Measurements are combined with the coefficient-space Two-Step procedure of Lee and Shin (arXiv:2309.01020), while a training-only decoder and regularization stabilize the adaptive coordinates. We derive a discrete error decomposition separating measurement, output-basis, and neural-approximation errors, together with a Barron-rate refinement. The framework is evaluated on the antiderivative operator, a non-normable locally convex input space, heterogeneous Darcy flow, a controlled operator, and fixed-time and time-evolving Navier-Stokes vorticity operators. In the heterogeneous Darcy problem, the functional models retain nearly resolution-independent errors of 5.5-5.6% on unseen grids, while in the controlled problem adaptive measurements reduce the mean error below 1.2%. For the fixed-time Navier-Stokes problem, the Adaptive Topological DeepONet is the most accurate DeepONet-based model, attaining a mean relative $L^2$ error of 1.685% +/- 0.017% using 128 functional coordinates. A comparably sized Fourier neural operator (FNO; arXiv:2010.08895) achieves the lower error 0.832% +/- 0.172%, but requires the full 64x64 input field, twice the training time, and 10.7x greater peak GPU memory. The formulation provides compact, interpretable, and discretization-portable coordinates in the continuous dual $V'$, including for non-normable input spaces.

We extend the $\U$-fixed-point and $\U$-startpoint framework for quasi-uniform spaces. We reformulate the original existence framework through the filter axioms and supply a two-sided contraction condition on a generating family of quasi-pseudometrics. For this condition we prove a Banach-style fixed-point theorem on $T_0$ bicomplete quasi-uniform spaces, with a Picard-type error bound; a Boyd–Wong companion and a stability estimate follow. The conjugate quasi-uniformity yields an endpoint version, a selector argument extends the result to multivalued maps and to common startpoints of commuting families, a Knaster–Tarski variant covers monotone selectors on order-complete spaces, and a direct quasi-Hausdorff route handles weakly contractive multivalued maps. Moreover, the $T_0$-quotient removes the $T_0$ assumption from the corresponding involution-based fixed-point theorem.

We start by rephrasing some well-known characterizations of fit and subfit locales in terms of closure and interior operators on the system of sublocales. Using these characterizations, we are able to extend the notions of fitness and subfitness to a more general setting. This approach not only allows us to obtain more general results, but also to present new properties of fitness and subfitness (such as a formula for the computation of supplements of sublocales). Using this approach we were also able to simplify the proofs of some well-known results.

We construct a bounded complete dcpo P whose maximal point space is homeomorphic to the set R of all reals equipped with the cocountable topology, thereby answering an open problem. Using this dcpo, we obtain a new complete lattice whose Scott space is non-sober.

Let X be a charming space with a Lindelof Sigma kernel Y, and let B be the closure of Y in X. We show that the question whether every charming space is a D-space can be reduced first to the dense-kernel case and then to a closed core. We define Obs_cc(Y,B) as the set of boundary points x in B minus Y such that, for every open neighborhood U of x in X, the intersection of U and Y is not countably compact, and let H be the closure of Obs_cc(Y,B) in B. We prove that the intersection of H with B minus Y is exactly Obs_cc(Y,B), and our main reduction theorem shows that X is a D-space if and only if H is. We also prove the following sufficient condition. If there is a closed set S contained in B minus Y with compact covering number less than the dominating number d, such that every point of B minus Y minus S has an open neighborhood U in X for which the intersection of U and Y is countably compact, then X is a D-space.

For each countable ordinal $\alpha\geq 2$, Filipów, Kowalczuk and Kwela introduced an ideal $\mathsf{conv}_\alpha$ on the countable compact ordinal space $\omega^\alpha+1$. Kowalczuk later proved that, for each countable limit ordinal $\lambda$, the ideal $\mathsf{conv}_{<\lambda}$ is the greatest lower bound of $\{\mathsf{conv}_\beta:\beta<\lambda\}$ in the Katětov order. At the first uncountable level, let $A\subseteq[2,\omega_1)$ be uncountable and let $D$ be a countable dense subset of $X_A=\prod_{\alpha\in A}(\omega^\alpha+1)$. The coordinate ideal $\mathsf{Conv}(A,D)$ on $D$ consists of those $B\subseteq D$ with $\pi_\alpha[B]\in\mathsf{conv}_\alpha$ for every $\alpha\in A$. For a pair $D\subseteq E$ of countable dense sets, call $\alpha$ non-small if $\pi_\alpha[E\setminus D]\notin\mathsf{conv}_\alpha$. In ZFC, if at most countably many coordinates are non-small, then $\mathsf{Conv}(A,D)\equiv_K\mathsf{Conv}(A,E)$. Under CH this countability bound is sharp: for every $A\subseteq[3,\omega_1)$ with $|A|=\aleph_1$, there are countable dense sets $D\subseteq D^*\subseteq X_A$ such that $\mathsf{Conv}(A,D^*)\leq_K\mathsf{Conv}(A,D)$ but $\mathsf{Conv}(A,D)\not\leq_K\mathsf{Conv}(A,D^*)$, and in particular $\mathsf{Conv}(A,D)$ and $\mathsf{Conv}(A,D^*)$ are not Katětov equivalent. The non-reduction is obtained, under CH, by diagonalizing along $\omega_1$ coordinates against the elements of $\omega^\omega$ that code retractions $D^*\to D$.

This paper completely classifies when the pro-$p$ completion of an amalgamated product has bounded generation, assuming its starting vertex groups are already boundedly generated. We first prove that the pro-$p$ completion of any abstract amalgam always naturally forms a proper pro-$p$ amalgam. Within this framework, we show that bounded generation is extremely rare. Specifically, the completed group is boundedly generated if and only if one of the vertex groups completely collapses into the edge group (meaning it has an index of 1), or in the highly specific case where $p=2$ and both vertex groups have an index of exactly 2 over the edge intersection.Additionally, we develop a new set of criteria to determine when the original vertex and edge groups map into the completion without collapsing. Finally, we provide a cautionary counterexample: for any given prime $p$, we construct an amalgam whose pro-$p$ completion is boundedly generated, but its full profinite completion is not. This demonstrates that possessing bounded generation at a single prime does not guarantee that the full profinite group will share this property.

In this paper, we give a co-sober space $X$ whose Smyth power space $P_S(X)$ is not co-sober, which gives a negative answer to an open problem.

Zhao and Ho use the irreducible open cover family respected with the Scott topology to introduce the SI-compactness. It has been proved that the SI-compactness is strictly stronger than compactness. In this paper, we prove that the product of finitely many SI-compact spaces is always SI-compact which gives a positive answer for the question posed by Zhao and Ho. Furthermore, we show that the category of all SI-compact spaces is not a reflective full subcategory of TOP.