Mathematics 205A Topology — I Course Notes Revised, Fall 2005
... Partial ordering of cardinalities Definition. If A and B are sets, we write |A| ≤ |B| if there is a 1–1 map from A to B. It follows immediately that this relation is transitive and reflexive, but the proof that it is symmetric is decidedly nontrivial: SCHRÖDER-BERNSTEIN THEOREM. If A and B are sets ...
... Partial ordering of cardinalities Definition. If A and B are sets, we write |A| ≤ |B| if there is a 1–1 map from A to B. It follows immediately that this relation is transitive and reflexive, but the proof that it is symmetric is decidedly nontrivial: SCHRÖDER-BERNSTEIN THEOREM. If A and B are sets ...
Open problems on countable dense homogeneity
... We do not know to whom this problem belongs. By the result of Kennedy [48], for continua this is equivalent to asking whether every 2-homogeneous metric continuum admits a homeomorphism that is the identity on some non-empty open set. On the other hand, there are even Baire SLH connected separable m ...
... We do not know to whom this problem belongs. By the result of Kennedy [48], for continua this is equivalent to asking whether every 2-homogeneous metric continuum admits a homeomorphism that is the identity on some non-empty open set. On the other hand, there are even Baire SLH connected separable m ...
Ordered Quotients and the Semilattice of Ordered
... of the ordered quotient space associated with a decomposition (only an ordered quotient space). Another definition for an ordered quotient space was introduced by McCartan [5] who gave a constructive definition for the order on the blocks of a decomposition. He defined [x] ≤ [y] iff for all x∗ ∈ [x] and ...
... of the ordered quotient space associated with a decomposition (only an ordered quotient space). Another definition for an ordered quotient space was introduced by McCartan [5] who gave a constructive definition for the order on the blocks of a decomposition. He defined [x] ≤ [y] iff for all x∗ ∈ [x] and ...