# Product-closed property of topological spaces

From Topospaces

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 metapropertyVIEW RELATED: topological space metaproperty satisfactions| topological space metaproperty dissatisfactions

## Contents

## Definition

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

### Weaker 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.