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We introduce the notion of an operation $\mathcal{P}$ and a $\mathcal{P}$-determined space. It is shown that a category is a coreflective full subcategory of $\bf Top$ if and only if it is equal to $\bf Top_{\mathcal{P}}$ for some idempotent and consistent operation $\mathcal{P}$, where $\bf Top_{\mathcal{P}}$ is the category of all $\mathcal{P}$-determined spaces. As concrete examples of $\mathcal{P}$-determined spaces, several classes of topological spaces determined by monotone convergence classes are investigated in detail. By the tool of $\mathbb{C}$-generated spaces, it is shown uniformly that categories of these examples of $\mathcal{P}$-determined spaces are all cartesian closed. Moreover, the exponential objects and categorical products of some categories in domain theory are shown to be closely related to those of $\bf DTop$, the category of directed spaces.

Powerdomains in domain theory plays an important role in modeling the semantics of nondeterministic functional programming languages.\ In this paper,\ we extend the notion of powerdomain to the category of directed spaces,\ which is equivalent to the notion of the\ $T_0$\ monotone-determined space\ \cite{EN2009}.\ We define the notion of upper,\ lower and convex powerspace of a directed space by the way of free algebras.\ We show that the upper,\ lower and convex powerspace over any directed space exist and give their concrete structures.\ Generally,\ the upper,\ lower and convex powerspaces of a directed spaces are different from the upper,\ lower and convex powerdomains of a dcpos endowed with the Scott topology and the observationally-induced upper and lower powerspaces introduced by Battenfeld and Sch\"{o}der in 2015. Keywords: powerdomain,\ directed lower powerspace of directed spaces,\ directed upper powerspace of directed spaces,\ directed convex powerspace of directed spaces,\ observationally-induced lower powerspace,\ observationally-induced lower powerspace, Comment: arXiv admin note: text overlap with arXiv:2203.04506

Probabilistic powerdomain in domain theory plays an important role in modeling the semantics of nondeterministic functional programming languages with probabilistic choice. In this paper, we extend the notion of powerdomain to directed spaces, which is equivalent to the notion of the T0 monotone-determined space [4]. We construct the probabilistic powerspace of the directed space, which is defined as a free directed space-cone. In addition, the relationships between our construction and classical probabilistic powerdomain are studied., Comment: 14 pages

Purpose: To assess whether a generative adversarial network (GAN) could synthesize realistic optical coherence tomography (OCT) images that satisfactorily serve as the educational images for retinal specialists and the training datasets for the classification of various retinal disorders using deep learning (DL). Methods: The GANs architecture was adopted to synthesis high-resolution OCT images training on a publicly available OCT dataset including urgent referrals (choroidal neovascularization and diabetic macular edema) and non-urgent referrals (normal and drusen). 400 real and synthetic OCT images were evaluated by 2 retinal specialists to assess image quality. We further trained 2 DL models on either real or synthetic datasets and compared the performance of urgent vs nonurgent referrals diagnosis tested on a local (1000 images from the public dataset) and clinical validation dataset (278 images from Shanghai Shibei Hospital). Results: The image quality of real vs synthetic OCT images was similar as assessed by 2 retinal specialists. The accuracy of discrimination as real vs synthetic OCT images was 59.50% for retinal specialist 1 and 53.67% for retinal specialist 2. For the local dataset, the DL model trained on real (DL_Model_R) and synthetic OCT images (DL_Model_S) had an area under the curve (AUC) of 0.99, and 0.98 respectively. For the clinical dataset, the AUC was 0.94 for DL_Model_R, 0.90 for DL_Model_S. Conclusions: The GAN-synthetic OCT images can be used by clinicians for educational purposes and developing DL algorithms. Translational Relevance: The medical image synthesis based on GANs is promising in human and machine to fulfill clinical tasks., Comment: submitted