F-Borel空间的复杂性与表示
Complexities and Representations of F-Borel Spaces
摘要：
研究了拓扑空间在不同紧化下的$\mathcal F$ -Borel复杂性.我们提供了一个简单的证明，一个空间可以有任意多个不同的复杂性在不同的紧化。我们还发展了$\mathcal F$ -Borel集的表示理论，并展示了如何应用这一理论来证明遗传Lindelöf空间的复杂性是绝对的（也就是说，它在每个紧化中都是相同的）。我们使用这些表示来表征特定类别的拓扑空间所能达到的复杂性。这提供了第一个结果的另一种证明，并暗示存在一个具有非绝对加法复杂性的空间。我们讨论的方法所使用的Talagrand构建的第一个例子的空间与非绝对复杂性，希望提供一个解释，这是更容易比原来的。我们还讨论了复杂性与局部复杂性的关系，并给出了如何构造类汞齐紧化。
英文摘要：
We investigate the $\mathcal F$-Borel complexity of topological spaces in their different compactifcations. We provide a simple proof of the fact that a space can have arbitrarily many different complexities in different compactifications. We also develop a theory of representations of $\mathcal F$-Borel sets, and show how to apply this theory to prove that the complexity of hereditarily Lindelöf spaces is absolute (that is, it is the same in every compactification). We use these representations to characterize the complexities attainable by a specific class of topological spaces. This provides an alternative proof of the first result, and implies the existence of a space with non-absolute additive complexity. We discuss the method used by Talagrand to construct the first example of a space with non-absolute complexity, hopefully providing an explanation which is more accessible than the original one. We also discuss the relation of complexity and local complexity, and show how to construct amalgamation-like compactifications.
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