论文标题
紧凑的许多开放套件的交叉点是打开的
Intersections of compactly many open sets are open
论文作者
论文摘要
根据定义，任何拓扑空间的有限的开放式集合的交集是开放的。 Nachbin观察到，更一般而言，紧凑的许多开放式集合是打开的。此外，Nachbin应用了这一点，以获取有关拓扑和其他地方紧凑型集的各种事实的优雅证明。一个简单的计算表明，Nachbin的观察结果相当于众所周知的事实，即如果空间$ x $紧凑，那么投影映射$ z \ times x \ to z $对于每个太空$ z $都关闭。众所周知，相反的是：如果一个空格$ x $具有投影$ z \ times x \ to z $的属性，则每个空间$ z $都关闭，则$ x $是紧凑的。我们将其重新制定为Nachbin观察的相反，并将其应用于以下方面的（旧）定理的进一步优雅证明。我们还提供了一个新的证明，即（重新重新制定）一个事实，即当且仅当投影映射$ z \ times x \ to z $都关闭时，每个太空$ z $都关闭了空间$ x $。这是通过各种方式概括的，以获得有关连续函数，正确地图，相对紧凑性和紧凑型空间的新结果。特别是，我们根据开放集的晶格的scott拓扑来给出对紧凑型空间类别中二进制产品的内在描述。
By definition, the intersection of finitely many open sets of any topological space is open. Nachbin observed that, more generally, the intersection of compactly many open sets is open. Moreover, Nachbin applied this to obtain elegant proofs of various facts concerning compact sets in topology and elsewhere. A simple calculation shows that Nachbin's observation amounts to the well known fact that if a space $X$ is compact, then the projection map $Z \times X \to Z$ is closed for every space $Z$. It is also well known that the converse holds: if a space $X$ has the property that the projection $Z \times X \to Z$ is closed for every space $Z$, then $X$ is compact. We reformulate this as a converse of Nachbin's observation, and apply this to obtain further elegant proofs of (old and new) theorems concerning compact sets. We also provide a new proof of (a reformulation of) the fact that a space $X$ is compact if and only if the projection map $Z \times X \to Z$ is closed for every space $Z$. This is generalized in various ways, to obtain new results about spaces of continuous functions, proper maps, relative compactness, and compactly generated spaces. In particular, we give an intrinsic description of the binary product in the category of compactly generated spaces in terms of the Scott topology of the lattice of open sets.
