论文标题
在结构化空间及其特性上
On structured spaces and their properties
论文作者
论文摘要
在本文中，我们引入了一种新型的拓扑空间，称为“结构化空间”，该空间在本地类似于各种代数结构。例如，这可能是有用的，可以在本地研究不能使用代数的工具在全球范围内赋予代数结构的空间。这些空间的定义将通过我们的主要结果之一，涉及“结构图”。这也将使我们对代数结构进行严格且明确的定义。在展示了一些自然出现在这种情况下的示例之后，我们研究了各种属性，并为这些新空间开发了一些理论。特别是，我们考虑分区（关于某种量度$μ$）。然后，我们证明了本文最重要的定理之一（定理4.1），该定理指出，在某些假设下，每个结构化空间都会引起晶格，相反，每个晶格都会诱导满足这种假设的结构化空间。我们与连接的空间有一些关系结束。
In this paper we introduce a new kind of topological space, called 'structured space', which locally resembles various kinds of algebraic structures. This can be useful, for instance, to locally study a space that cannot be globally endowed with an algebraic structure using tools from algebra. The definition of these spaces will be made more precise via one of our main result, which involves the 'structure map'. This will also lead us to a rigorous and unambiguous definition of algebraic structure. After showing some examples which naturally arise in this context, we study various properties and develop some theory for these new spaces; in particular, we consider partitions (with respect to some measure $μ$). We then prove one of the most important Theorem of this paper (Theorem 4.1), which states that every structured space, under some assumptions, induces a lattice, and conversely every lattice induces a structured space satisfing such hypothesis. We conclude with some relations with connected spaces.
