In this paper we extend the notion of a clopen subset of a topological space to that of a clopen subobject in a topological category. We relate clopen subobjects to the concept of normalized topological category via a theorem which, in turn, shows why certain subsets must be clopen in a topological space. We employ clopen subobjects to extend the topological notion of connectedness to topological categories. Examples of clopen subobjects and connected objects in several topological categories are given.

中文翻译：

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Clopen对象，连接的对象和规范化的拓扑类别

在本文中，我们将拓扑空间的clopen子集的概念扩展到拓扑类别中的clopen子对象的概念。通过一个定理，我们将clopen子对象与归一化拓扑类别的概念相关联，该定理又表明了为什么某些子集必须在拓扑空间中是clopen的。我们使用clopen子对象将连通性的拓扑概念扩展到拓扑类别。给出了几个拓扑类别中的clopen子对象和连接对象的示例。