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A combinatorial statement concerning ideals of countable subsets of ω is introduced and proved to be consistent with the Continuum Hypothesis. This statement implies the Suslin Hypothesis, that all (ω, ω*)-gaps are Hausdorff, and that every coherent sequence on ω either almost includes or is orthogonal to some uncountable subset of ω.

Partitioning bases of topological spaces

Dániel T. Soukup, Lajos Soukup (2014)

Commentationes Mathematicae Universitatis Carolinae

We investigate whether an arbitrary base for a dense-in-itself topological space can be partitioned into two bases. We prove that every base for a $T_{3}$ Lindelöf topology can be partitioned into two bases while there exists a consistent example of a first-countable, 0-dimensional, Hausdorff space of size $2^{ω}$ and weight $ω_{1}$ which admits a point countable base without a partition to two bases.

Partitioning spaces into homeomorphic rigid parts

Jan van Mill, Evert Wattel (1985)

Colloquium Mathematicae

Partitions of compact Hausdorff spaces

Gary Gruenhage (1993)

Fundamenta Mathematicae

Under the assumption that the real line cannot be covered by $ω_{1}$ -many nowhere dense sets, it is shown that (a) no Čech-complete space can be partitioned into $ω_{1}$ -many closed nowhere dense sets; (b) no Hausdorff continuum can be partitioned into $ω_{1}$ -many closed sets; and (c) no compact Hausdorff space can be partitioned into $ω_{1}$ -many closed $G_{δ}$ -sets.

Partitions of pairs of reals

Alan Taylor (1978)

Fundamenta Mathematicae

Pasting topological spaces at one point

Ali Rezaei Aliabad (2006)

Czechoslovak Mathematical Journal

Let ${X_{α}}_{α \in Λ}$ be a family of topological spaces and $x_{α} \in X_{α}$ , for every $α \in Λ$ . Suppose $X$ is the quotient space of the disjoint union of $X_{α}$ ’s by identifying $x_{α}$ ’s as one point $σ$ . We try to characterize ideals of $C (X)$ according to the same ideals of $C (X_{α})$ ’s. In addition we generalize the concept of rank of a point, see [9], and then answer the following two algebraic questions. Let $m$ be an infinite cardinal. (1) Is there any ring $R$ and $I$ an ideal in $R$ such that $I$ is an irreducible intersection of $m$ prime ideals? (2) Is there...

PCA sets and convexity

R. Kaufman (2000)

Fundamenta Mathematicae

Three sets occurring in functional analysis are shown to be of class PCA (also called $Σ_{2}^{1}$ ) and to be exactly of that class. The definition of each set is close to the usual objects of modern analysis, but some subtlety causes the sets to have a greater complexity than expected. Recent work in a similar direction is in [1, 2, 10, 11, 12].

Peano compactifications and property S metric spaces.

Dickman, Raymond F.jun. (1980)

International Journal of Mathematics and Mathematical Sciences

Peculiar behavior of connected locales

Igor Kříž, Aleš Pultr (1989)

Cahiers de Topologie et Géométrie Différentielle Catégoriques

P-embedding and product spaces

Kiiti Morita, Takao Hoshina (1976)

Fundamenta Mathematicae

Perfect compactifications of frames

Dharmanand Baboolal (2011)

Czechoslovak Mathematical Journal

Perfect compactifications of frames are introduced. It is shown that the Stone-Čech compactification is an example of such a compactification. We also introduce rim-compact frames and for such frames we define its Freudenthal compactification, another example of a perfect compactification. The remainder of a rim-compact frame in its Freudenthal compactification is shown to be zero-dimensional. It is shown that with the assumption of the Boolean Ultrafilter Theorem the Freudenthal compactification...

Perfect compactifications of functions

Giorgio Nordo, Boris A. Pasynkov (2000)

Commentationes Mathematicae Universitatis Carolinae

We prove that the maximal Hausdorff compactification $χ f$ of a $T_{2}$ -compactifiable mapping $f$ and the maximal Tychonoff compactification $β f$ of a Tychonoff mapping $f$ (see [P]) are perfect. This allows us to give a characterization of all perfect Hausdorff (respectively, all perfect Tychonoff) compactifications of a $T_{2}$ -compactifiable (respectively, of a Tychonoff) mapping, which is a generalization of two results of Skljarenko [S] for the Hausdorff compactifications of Tychonoff spaces.

Perfect mappings in topological groups, cross-complementary subsets and quotients

Aleksander V. Arhangel'skii (2003)

Commentationes Mathematicae Universitatis Carolinae

The following general question is considered. Suppose that $G$ is a topological group, and $F$ , $M$ are subspaces of $G$ such that $G = M F$ . Under these general assumptions, how are the properties of $F$ and $M$ related to the properties of $G$ ? For example, it is observed that if $M$ is closed metrizable and $F$ is compact, then $G$ is a paracompact $p$ -space. Furthermore, if $M$ is closed and first countable, $F$ is a first countable compactum, and $F M = G$ , then $G$ is also metrizable. Several other results of this kind are obtained....

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