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Product-closed property of topological spaces

Definition

Suppose \alpha is a property of topological spaces, i.e., for any topological space X, X either satisfies \alpha or does not satisfy \alpha. Then, we say that \alpha is a product-closed property of topological spaces if for any (possibly finite, possibly infinite) collection of topological spaces X_i, i \in I, all of which satisfy \alpha, the product space \prod_{i \in I} X_i, endowed with the product topology, also satisfies \alpha.

Relation with other metaproperties

Weaker metaproperties

Other related 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.