In mathematics, S-space is a regular topological space that is hereditarily separable but is not a
Lindelöf space. L-space is a regular topological space that is hereditarily
Lindelöf but not separable. A space is separable if it has a countable dense set and hereditarily separable if every subspace is separable.
It had been believed for a long time that S-space problem and L-space problem are dual, i.e. if there is an S-space in some model of set theory then there is an L-space in the same model and vice versa – which is not true.
It was shown in the early 1980s that the existence of S-space is independent of the usual axioms of
ZFC. This means that to prove the existence of an S-space or to prove the non-existence of S-space, we need to assume axioms beyond those of
ZFC. The L-space problem (whether an L-space can exist without assuming additional set-theoretic assumptions beyond those of
ZFC) was not resolved until recently.
Todorcevic proved that under
PFA there are no S-spaces. This means that every regular
hereditarily separable space is
Lindelöf. For some time, it was believed the L-space problem would have a similar solution (that its existence would be independent of
ZFC).
Todorcevic showed that there is a model of set theory with
Martin's axiom
In the mathematical field of set theory, Martin's axiom, introduced by Donald A. Martin and Robert M. Solovay, is a statement that is independent of the usual axioms of ZFC set theory. It is implied by the continuum hypothesis, but it is consist ...
where there is an L-space but there are no S-spaces. Further,
Todorcevic found a compact S-space from a
Cohen real.
In 2005,
Moore solved the L-space problem by constructing an L-space without assuming additional axioms and by combining
Todorcevic's
rho functions with
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mat ...
.
Sources
* K. P. Hart, Juniti Nagata, J.E. Vaughan: ''Encyclopedia of General Topology'', Elsevier, 2003 ,
* Stevo Todorcevic: "Partition problems in topology" (Chapter 2, 5, 6, and 9), ''Contemporary Mathematics'', 1989: Volume 84 ,
* Justin Tatch Moore: "A Solution to the L Space Problem", ''Journal of the American Mathematical Society'', Volume 19, pages 717–736, 2006
{{topology-stub
General topology
Topological spaces