Cosmic dimensions
Martin's Axiom for $σ$-centered partial orders implies that there is a cosmic space with non-coinciding dimensions.
math.GN↗
arXiv subjects
Alan Dow
Publications and source records attributed to Alan Dow.
Applications of another characterization of betaN\N
Steprans provided a characterization of betaN\N in the aleph_2-Cohen model that is much in the spirit of Parovicenko's characterization of this space under CH. A variety of the topological results established in the Cohen model can be deduced directly from the properties of betaN\N or P(N)/fin that feature in Steprans' result.
math.GN↗
Hereditary indecomposability and the Intermediate Value Theorem
We show that hereditarily indecomposable spaces can be characterized by a special instance of the Intermediate Value Theorem in their rings of continuous functions.
math.GN↗
The measure algebra does not always embed
The Open Colouring Axiom implies that the measure algebra cannot be embedded into P(N)/fin. We also discuss the errors in previous results on the embeddability of the measure algebra.
math.LO↗
A Universal Continuum of Weight aleph
We prove that every continuum of weight aleph_1 is a continuous image of the Cech-Stone-remainder R^* of the real line. It follows that under CH the remainder of the half line [0,infty) is universal among the continua of weight c --- universal in the `mapping onto' sense. We complement this result by showing that 1) under MA every continuum of weight less than c is a continuous image of R^* 2) in the Cohen model the long segment of length omega_2+1 is not a continuous image of R^*, and 3) PFA implies that I_u is not a continuous image of R^*, whenever u is a c-saturated ultrafilter. We also show that a universal continuum can be gotten from a c-saturated ultrafilter on omega and that it is consistent that there is no universal continuum of weight c.
math.GN↗
Čech-Stone remainders of spaces that look like $[0,\infty)$
We show that many spaces that look like the half~line~$\halfline=[0,\infty)$ have, under~$\CH$, a Čech-Stone-remainder that is homeomorphic to~$\Hstar$. We also show that $\CH$ is equivalent to the statement that all standard subcontinua of~$\Hstar$ are homeomorphic. The proofs use Model-theoretic tools like reduced products and elementary equivalence; rather than constructing homeomorphisms we show that the spaces in question have isomorphic bases for the closed sets.
math.LO↗