Coarse topologies on the real line

摘要

Let c = | R | denote the cardinality of the continuum and let η denote the Euclidean topology on R . Let L denote the family of all Hausdor. topologies τ on R with τ . η . Let L 1 resp. L 2 resp. L 3 denote the family of all τ ∈ L where ( R , τ ) is completely normal resp. second countable resp. not regular. Trivially, L 1 ∩ L 3 = . and |L i | ≤ |L| ≤ 2 c and |L 2 | ≤ c. For τ ∈ L the space ( R , τ) is metrizable if and only if τ ∈ L 1 ∩ L 2 . We show that, up to homeomorphism, both L 1 and L 3 contain precisely 2 c topologies and L 2 contains precisely c completely metrizable topologies. For 2 c non-homeomorphic topologies τ ∈ L 1 the space ( R , τ ) is Baire, but there are also 2 c non-homeomorphic topologies τ ∈ L 1 and c non-homeomorphic topologies τ ∈ L 1 ∩ L 2 where ( R , τ ) is of first category. Furthermore, we investigate the complete lattice L 0 of all topologies τ ∈ L such that τ and η coincide on R {0}. In the lattice L 0 we find 2 c (non-homeomorphic) immediate predecessors of the maximum η , whereas the minimum of L 0 is a compact topology without immediate successors in L 0 . We construct chains of homeomorphic topologies in L 0 ∩ L 1 ∩ L 2 and in L 0 ∩ L 2 ∩ L 3 and in L 0 ∩ (L 1 L 2 ) and in L 0 ∩ (L 3 L 2 ) such that the length of each chain is c (and hence maximal). We also track down a chain in L 0 of length 2 λ where λ is the smallest cardinal number κ with 2 κ > c.