Alan Dow’s research while affiliated with Indiana University Bloomington and other places

The normal Moore space conjecture asserts that normal Moore spaces are metrizable. Nyikos has proven the consistency (from the existence of a strongly compact cardinal) of the conjecture holding and Fleissner has proven that at least a measurable cardinal is needed to prove the consistency. Although extremely elegant, Nyikos' proof relies on Kunen's proof of the consistency of the product measure extension axiom and does not lend itself to other applications. In this paper we first present the groundwork for iterated forcing and reflection type proofs from the assumption of a supercompact cardinal. We then use this technology to give a proof of the normal Moore space conjecture as well as several other similar results which use a variation of the proof.

In this article, it is shown to be consistent that is not C∗-embedded in βω2. It is also shown that if the existence of a weakly compact cardinal is consistent with ZFC, then so is the negation of the previous statement. Furthermore, both results are independent of CH.

We determine those regular cardinals κ with the property that for each increasing κ-chain of first countable spaces there is a compatible first countable topology on the union of the chain. AssumingV=L any such κ must be weakly compact. It is relatively consistent with a supercompact cardinal that each κ>w 1 has the property. The proofs exploit the connection with interesting families of integer-valued functions.

In the model obtained by adding $\omega_2$ side-by-side Sacks reals to a model of CH, there is a separable nonpseudocompact space with no remote points. To prove this it is also shown that in this model the countable box product of Cantor sets contains a subspace of size $\omega_2$ such that every uncountable subset has density $\omega_1$. Furthermore assuming the existence of a measurable cardinal $\kappa$ with $2^\kappa = \kappa^+$, a space X is produced with no isolated points but with remote points in $vX - X$. It is also shown that a pseudocompact space does not have remote points.

In the model obtained by adding ω 2 {\omega _2} side-by-side Sacks reals to a model of C H {\mathbf {CH}} , there is a separable nonpseudocompact space with no remote points. To prove this it is also shown that in this model the countable box product of Cantor sets contains a subspace of size ω 2 {\omega _2} such that every uncountable subset has density ω 1 {\omega _1} . Furthermore assuming the existence of a measurable cardinal κ \kappa with 2 κ = κ + {2^\kappa } = {\kappa ^ + } , a space X X is produced with no isolated points but with remote points in υ X − X \upsilon X - X . It is also shown that a pseudocompact space does not have remote points.

Examples of products with remote points and counterexamples of products without remote points are given. The paradoxical behavior of remote points with respect to products is exhibited. Also, an example is given of spaces X and Y, where neither X nor Y has a σ-locally finite π-base, but X×Y does.

It is shown that a product of metric spaces has remote points if and only if it is noncompact. Within the proof, infinitary combinatoric methods are developed and implemented to create an effective strategy for a two player game. Many new examples are given of nonpseudocompact products which have the property that the nonhomogeneity of the remainder can be demonstrated by an explicit pair of points.