arXiv subjects
Publications and source records attributed to Paolo Lipparini.
We extend a theorem by Juhász and Szentmiklóssy to notions related to pseudocompactness. We also allow the case when one of the cardinals under consideration is singular. We give an application to the study of decomposable ultrafilters: if κ is singular, D is a uniform ultrafilter over κ^+, and D' is a uniform ultrafilter over cf κ, then D' \times D is κ-decomposable.
We characterize the compactness properties of the product of λ copies of the space ω with the discrete topology, dealing in particular with the case λ singular, using regular and uniform ultrafilters, infinitary languages and nonstandard elements. We also deal with products of uncountable regular cardinals with the order topology.
We characterize exactly the compactness properties of the product of κ copies of the space ω with the discrete topology. The characterization involves uniform ultrafilters, infinitary languages, and the existence of nonstandard elements in elementary estensions. We also have results involving products of possibly uncountable regular cardinals.
If $\mathcal P$ is a family of filters over some set $I$, a topological space $X$ is \emph{sequencewise $\mathcal P$-\brfrt compact} if, for every $I$-indexed sequence of elements of $X$, there is $F \in \mathcal P$ such that the sequence has an $F$-limit point. Countable compactness, sequential compactness, initial $κ$-compactness, $[ λ,μ]$-compactness, the Menger and Rothberger properties can all be expressed in terms of sequencewise $\mathcal P$-compactness, for appropriate choices of $\mathcal P$. We show that sequencewise $\mathcal P$-compactness is preserved under taking products if and only if there is a filter $F \in \mathcal P$ such that sequencewise $\mathcal P$-compactness is equivalent to $F$-compactness. If this is the case, and there exists a sequencewise $\mathcal P$-compact $T_1$ topological space with more than one point, then $F$ is necessarily an ultrafilter. The particular cases of sequential compactness and of $[ λ,μ]$-compactness are analyzed in detail.
We show that if $μ\leq \cf λ$ and $λ$ is a strong limit singular cardinal, then $[μ, λ]$-compactness is productive if and only if either $μ= ω$, or $μ$ is $λ$-strongly compact.
We characterize various Menger/Rothberger related properties by means of ultrafilter convergence, and discuss their behavior with respect to products.
We show that the product of any number of sequentially pseudocompact topological spaces is still sequentially pseudocompact. The definition of sequential pseudocompactness can be given in (at least) two ways: we show their equivalence. Some of the results of the present note already appeared in A. Dow, J. R. Porter, R. M. Stephenson, R. G. Woods, Spaces whose pseudocompact subspaces are closed subsets, Appl. Gen. Topol. 5 (2004), 243-264.
An identity s=t is linear if each variable occurs at most once in each of the terms s and t. Let T be a tolerance relation of an algebra A in a variety defined by a set of linear identities. We prove that there exist an algebra B in the same variety and a congruence Theta of B such that a homomorphism from B onto A maps Theta onto T.
We discuss two possible ways of representing tolerances: first, as a homomorphic image of some congruence; second, as the relational composition of some compatible relation with its converse. The second way is independent from the variety under consideration, while the first way is variety-dependent. The relationships between these two kinds of representations are clarified. As an application, we show that any tolerance on some lattice L is the image of some congruence on a subalgebra of L $\times$ L. This is related to recent results by G. Czédli and E. W. Kiss.
We prove that, for an arbitrary topological space $X$, the following two conditions are equivalent: (a) Every open cover of $X$ has a finite subset with dense union (b) $X$ is $D$-pseudocompact, for every ultrafilter $D$. Locally, our result asserts that if $X$ is weakly initially $λ$-compact, and $2^ μ\leq λ$, then $X$ is $D$-\brfrt pseudocompact, for every ultrafilter $D$ over any set of cardinality $ \leq μ$. As a consequence, if $2^ μ\leq λ$, then the product of any family of weakly initially $λ$-compact spaces is weakly initially $μ$-compact.
We characterize the situations in which certain accumulation properties of topological spaces are preserved under taking products.
We introduce a new covering property, defined in terms of order types of sequences of open sets, rather than in terms of cardinalities of families. The most general form of this compactness notion depends on two ordinal parameters. In the particular case when the parameters are cardinal numbers, we get back a classical notion. Generalized to ordinal numbers, this notion turns out to behave in a much more varied way. We present many examples of spaces satisfying the very same cardinal compactness properties, but with a broad range of distinct behaviors, with respect to ordinal compactness. A much more refined theory is obtained for $T_1$ spaces, in comparison with arbitrary topological spaces. The notion of ordinal compactness becomes partly trivial for spaces of small cardinality.
We discuss the connection between various orders on the class of all the ultrafilters and certain compactness properties of abstract logics and of topological spaces. We present a model theoretical characterization of Comfort order. We introduce a new order motivated by considerations in abstract model theory. For each of the above orders, we show that if $E$ is a $(λ, λ)$-regular ultrafilter, and $D$ is not $(λ, λ)$-regular, then $E \not \leq D$. Many problems are stated.
The statement in the title solves a problem raised by T. Retta. We also present a variation of the result in terms of $[ μ,κ]$-compactness.
We introduce a covering notion depending on two cardinals, which we call $\mathcal O $-$ [ μ, λ]$-compactness, and which encompasses both pseudocompactness and many other generalizations of pseudocompactness. For Tychonoff spaces, pseudocompactness turns out to be equivalent to $\mathcal O $-$ [ ω, ω ]$-compactness. We provide several characterizations of $\mathcal O $-$ [ μ, λ]$-compactness, and we discuss its connection with $D$-pseudocompactness, for $D$ an ultrafilter. We analyze the behaviour of the above notions with respect to products. Finally, we show that our results hold in a more general framework, in which compactness properties are defined relative to an arbitrary family of subsets of some topological space $X$.
We discuss some notions of compactness and convergence relative to a specified family F of subsets of some topological space X. The two most interesting particular cases of our construction appear to be the following ones. (1) The case in which F is the family of all singletons of X, in which case we get back the more usual notions. (2) The case in which F is the family of all nonempty open subsets of X, in which case we get notions related to pseudocompactness. A large part of the results in this note are known in particular case (1); the results are, in general, new in case (2). As an example, we characterize those spaces which are D-pseudocompact, for some ultrafilter D uniform over $λ$.
We discuss the existence of complete accumulation points of sequences in products of topological spaces. Then we collect and generalize many of the results proved in Parts I, II and IV. The present Part VI is complementary to Part V to the effect that here we deal, say, with uniformity, complete accumulation points and $ κ$-$(λ)$-compactness, rather than with regularity, $[ λ, μ]$-compactness and $ κ$-$ (λ, μ)$-compactness. Of course, if we restrict ourselves to regular cardinals, Parts V (for $ λ= μ$) and Part VI essentially coincide.
We generalize to the relations $(λ, μ) \stackrelκ{\Rightarrow} (λ', μ')$ and $\alm (λ, μ) \stackrelκ{\Rightarrow} \alm (λ', μ')$ some results obtained in Parts II and IV. We also present a multi-cardinal version.