How a Change to Topology's Union Axiom Affects Continuity

Countable products and countable sums of compact metrizable spaces in the absence of the Axiom of Choice

The main aim of the article is to show, in the absence of the Axiom of Choice, relationships between the following, independent of $\mathbf{ZF}$, statements: "Every countable product of compact metrizable spaces is separable (respectively, compact)" and "Every countable product of compact metrizable spaces is metrizable". Statements related to the above-mentioned ones are also studied. Permutation models (among them new ones) are shown in which a countable sum (also a countable product) of metrizable spaces need not be metrizable, countable unions of countable sets are countable and there is a…

软均匀空间与软均匀连续性：诱导拓扑、分离性与紧致性类型结果

Soft Uniform Spaces and Soft Uniform Continuity: Induced Topologies, Separation, and Compactness-Type Results

本文研究了软均匀空间和软均匀连续性，探讨了诱导拓扑、分离性和紧致性等性质，并证明了软海恩-康托尔定理及软紧致性蕴含软总bounded性和软完备性。

Locally compact, $ω_1$-compact spaces

An $ω_1$-compact space is a space in which every closed discrete subspace is countable. We give various general conditions under which a locally compact, $ω_1$-compact space is $σ$-countably compact, i.e., the union of countably many countably compact spaces. These conditions involve very elementary properties. Many results shown here are independent of the usual (ZFC) axioms of set theory, and the consistency of some may involve large cardinals. For example, it is independent of the ZFC axioms whether every locally compact, $ω_1$-compact space of cardinality $\aleph_1$ is $σ$-countably compac…