Product-closed property of topological spaces
This article defines a metaproperty of topological spaces: a property that can be evaluated to true/false for any property of topological spaces
View a complete list of metaproperties of topological spaces
View topological space properties satisfying this metaproperty| View topological space properties dissatisfying this metaproperty
VIEW RELATED: topological space metaproperty satisfactions| topological space metaproperty dissatisfactions
Suppose is a property of topological spaces, i.e., for any topological space , either satisfies or does not satisfy . Then, we say that is a product-closed property of topological spaces if for any (possibly finite, possibly infinite) collection of topological spaces , all of which satisfy , the product space , endowed with the product topology, also satisfies .
Relation with other metaproperties
To understand the relation between these concepts:
- Examples of properties of topological spaces that are both product-closed and box product-closed: T1 space, Hausdorff space
- Examples of properties that are box product-closed but not product-closed: discrete space
- Examples of properties that are product-closed but not box product-closed: compact space, connected space. Basically this list includes properties that are about smallness or intimacy of some sort. The box topology makes separation easier.