Locally upper bounded poset-valued maps and stratifiable spaces

Set-valued maps and some generalized metric spaces

To give characterizations of monotonically countably paracompact spaces with set-valued maps, Yamazaki [22] introduced the notion of strictly increasing closed cover of a topological space with which the boundedness of a set-valued map was defined. In this paper, we show that most of generalized metric spaces such as stratifiable spaces, semi-metrizable spaces can be characterized with set-valued maps with values into the family of all closed nonempty subsets of a space which has a strictly increasing closed cover. Moreover, as an application, we use the results obtained to give characterizati…

通过 $L$-sobriety 研究 frame 值论域的拓扑表示

Topological representations for frame-valued domains via $L$-sobriety

本文以 frame $L$ 为真值表，引入局部超紧及强局部超紧 $L$-拓扑空间，借助 $L$-sobriety 表示连续与代数 $L$-dcpo，并建立相关范畴同构及定向完备化应用。

Morse hyper-graphs and abstract weak orbit spaces of semi-decompositions

We introduce topological invariants of semi-decompositions (e.g. filtrations, semi-group actions, multi-valued dynamical systems, combinatorial dynamical systems) on a topological space to analyze semi-decompositions from a dynamical systems point of view. In fact, we construct Morse hyper-graphs and abstract weak elements of semi-decompositions. Moreover, the Morse hyper-graphs of the set of sublevel sets of a Morse function of a compact manifold is the Reeb graph of such function as abstract multi-graphs. The abstract weak element space for a simplicial complex is the face poset. The abstrac…