Stationary Sets Trees and Continuums

Stevo Todorčević

Publications de l'Institut Mathématique, _N_S_29 (1981) no. 43, p. 249 .

Résumé

The Suslin problem proposed in 1920 by M. Suslin has been very stimulating. The first advance on the problem was made by Đ. Kurepa in [K1] where he proved the equivalence of the existence of Suslin continuum ($\equiv$ a non-separable linearly ordered continuum with no uncountable family of disjoint open intervals) and Suslin tree ($\equiv$ an uncountable tree with no uncountable chains nor antichains). Since the construction of either the continuum or the tree seemed to be very hard, the above equivalence suggested the construction of an uncountable tree with countable levels and with no uncountable chains. Surprisingly, such a construction was indeed possible; this was done by N. Aronszajn in [K1; p 96]. But the above equivalence (in fact, its proof) also suggested the construction of a linearly ordered first countable continuum which has no dense set equal to the union of countably many discrete subspaces (see [K2]). We shall give such a construction in this paper.