# Lindelöf space

In mathematics, a **Lindelöf space**^{[1]}^{[2]} is a topological space in which every open cover has a countable subcover. The Lindelöf property is a weakening of the more commonly used notion of *compactness*, which requires the existence of a *finite* subcover.

A **hereditarily Lindelöf space**^{[3]} is a topological space such that every subspace of it is Lindelöf. Such a space is sometimes called **strongly Lindelöf**, but confusingly that terminology is sometimes used with an altogether different meaning.^{[4]}
The term *hereditarily Lindelöf* is more common and unambiguous.

Lindelöf spaces are named after the Finnish mathematician Ernst Leonard Lindelöf.

The thing to notice here is that each point on the antidiagonal is contained in exactly one set of the covering, so all these sets are needed.