In mathematics, a Lindelöf space 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 **** 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. The term hereditarily Lindelöf is more common and unambiguous. Lindelöf spaces are named after the Finnish mathematician Ernst Leonard Lindelöf.

Properties of Lindelöf spaces

Properties of hereditarily Lindelöf spaces

Example: the Sorgenfrey plane is not Lindelöf

The product of Lindelöf spaces is not necessarily Lindelöf. The usual example of this is the Sorgenfrey plane \mathbb{S}, which is the product of the real line \Reals 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. The antidiagonal of \mathbb{S} is the set of points (x, y) such that x + y = 0. Consider the open covering of \mathbb{S} 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. Another way to see that S is not Lindelöf is to note that the antidiagonal defines a closed and uncountable discrete subspace of S. This subspace is not Lindelöf, and so the whole space cannot be Lindelöf either (as closed subspaces of Lindelöf spaces are also Lindelöf).

Generalisation

The following definition generalises the definitions of compact and Lindelöf: a topological space is \kappa-compact (or \kappa-Lindelöf), where \kappa is any cardinal, if every open cover has a subcover of cardinality strictly less than \kappa. Compact is then \aleph_0-compact and Lindelöf is then \aleph_1-compact. The , or Lindelöf number l(X), is the smallest cardinal \kappa such that every open cover of the space X has a subcover of size at most \kappa. In this notation, X is Lindelöf if 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 \kappa such that every open cover of the space X has a subcover of size strictly less than \kappa. In this latter (and less used) sense the Lindelöf number is the smallest cardinal \kappa such that a topological space X is \kappa-compact. This notion is sometimes also called the of the space X.