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 usual example of this is the Sorgenfrey plane
which is the product of the real line
under the half-open interval topology with itself.
Open sets in the Sorgenfrey plane are unions of half-open rectangles that include the south and west edges and omit the north and east edges, including the northwest, northeast, and southeast corners.
which consists of: The thing to notice here is that each point on the antidiagonal is contained in exactly one set of the covering, so all the (uncountably many) sets of item (2) above are needed.
is not Lindelöf is to note that the antidiagonal defines a closed and uncountable discrete subspace of
such that every open cover of the space
The Lindelöf number as defined above does not distinguish between compact spaces and Lindelöf non-compact spaces.
Some authors gave the name Lindelöf number to a different notion: the smallest cardinal
such that every open cover of the space
has a subcover of size strictly less than
[17] In this latter (and less used) sense the Lindelöf number is the smallest cardinal
This notion is sometimes also called the compactness degree of the space