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Almost Disjoint Families and Topology

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Recent Progress in General Topology III

Abstract

An infinite family \({\fancyscript{A}}\subset {\fancyscript{P}}(\omega )\) is almost disjoint (AD) if the intersection of any two distinct elements of \({\fancyscript{A}}\) is finite. We survey recent results on almost disjoint families, concentrating on their applications to topology.

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Notes

  1. 1.

    Recall that (‘stick’) is the following weakening of \(\mathsf {CH}\): There is a family \({\fancyscript{S}}\subseteq [\omega _1]^\omega \) of size \(\aleph _1\) such that every uncountable subset of \(\omega _1\) contains an element of \({\fancyscript{S}}\).

  2. 2.

    A space \(X\) has property (a) if for every open cover \(\fancyscript{U}\) of \(X\) and a dense set \(D\subseteq X\) there is a closed discrete \(F\subseteq D\) such that \(st(F,\fancyscript{U})=X\).

  3. 3.

    If \(P_I\) is a proper forcing then it has CRN if for every Borel function \(f:B\rightarrow 2^\omega \) with an \(I\)-positive Borel domain \(B\) there is an \(I\)-positive Borel set \(C\subseteq B\) such that \(f\upharpoonright {C}\) is continuous.

  4. 4.

    A Banach space \(E\) is weak-norm-perfect if every norm-open subset of \(E\) is \(F_\sigma \) in the weak topology.

  5. 5.

    A separable metric space \(X\) is called a \(\gamma \) -set if every \(\omega \)-cover of \(X\) has a \(\gamma \)-subcover. An open cover \(\fancyscript{U}=\{U_n: n\in \omega \}\) is an \(\omega \) -cover if for every finite \(F\subseteq X\) there is a \(n\in \omega \) such that \(F\subseteq U_n\); \(\fancyscript{U}=\{U_n: n\in \omega \}\) is a \(\gamma \) -cover if for every \(x\in X\) and for all but finitely many \(n\in \omega \), \(x\in U_n\).

  6. 6.

    A filter \(\fancyscript{F}\) on \(\omega \) is a FUF-filter if given a family \(\fancyscript{H}\subseteq [\omega ]^{<\omega }\setminus \{\emptyset \}\) such that every element of \(\fancyscript{F}\) contains an element of \(\fancyscript{H}\) there is a sequence \(\{a_n: n\in \omega \}\subseteq \fancyscript{H}\) such that every element of \(\fancyscript{F}\) contains all but finitely many \(a_n\)’s. Reznichenko and Sipacheva in [137] noted that a FUF-filter (short for Fréchet-Urysohn for finite sets) produces a group topology on the Boolean group \([\omega ]^{<\omega }\) which is always Fréchet and is metrizable if and only if the filter has countable character.

  7. 7.

    Recall that a set of reals \(X\) is a \(\sigma \) -set if every \(G_\delta \) subset of \(X\) is \(F_\sigma \), and \(X\) is concentrated on a countable set \(D\subseteq X\) if every open set containing \(D\) contains all but countably many elements of \(X\).

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Acknowledgments

The author gratefully acknowledges support from a PAPIIT grant 102311. He wishes to thank profesor Alan Dow for patiently explaining to him Shelah’s proof of the existence of a completely separable MAD family and to profesor Piotr Koszmider for his help with section 9. He also wants to thank to Petr Simon, Osvaldo Guzmán, Rodrigo Hernández, Carlos Martínez and Ariet Ramos for commenting on preliminary versions of the paper.

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  1. Centro de Ciencias Matemáticas, Universidad Nacional Autónoma de México, Morelia, Mexico

    Michael Hrušák

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  1. Faculteit EWI, Technische Universiteit, Delft, The Netherlands

    K.P. Hart

  2. Faculty of Sciences, VU University, Amsterdam, The Netherlands

    J. van Mill

  3. Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic

    P. Simon

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Hrušák, M. (2014). Almost Disjoint Families and Topology. In: Hart, K., van Mill, J., Simon, P. (eds) Recent Progress in General Topology III. Atlantis Press, Paris. https://doi.org/10.2991/978-94-6239-024-9_14

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