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A topology generated by eventually different functions. (English) Zbl 0883.03033
The topology \(\mathcal E\) on \(\omega^\omega\) mentioned in the title is generated by the sets \(U_{s,F}=\{x\in\omega^\omega: s\subseteq x\mathrel{\&} (\forall f\in F)(\forall i\geq n)\;f(i)\neq x(i)\}\) for \(s\in2^n\), \(n\in\omega\), and finite sets \(F\subseteq\omega^\omega\). The space \((\omega^\omega,{\mathcal E})\) is a Tikhonov c.c.c. space in which no open set is meager. Cardinal coefficients for the ideal of meager sets \({\mathcal I}_{\mathcal E}\) are estimated by cardinals of Cichoń’s diagram and some applications of Martin’s axiom and anti-Martin’s axiom are obtained. It is proved that the additivity of \({\mathcal I}_{\mathcal E}\) is \(\omega_1\) and the minimal size of a base of \({\mathcal I}_{\mathcal E}\) is \(\mathfrak c\).

MSC:
03E15 Descriptive set theory
54H05 Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
03E40 Other aspects of forcing and Boolean-valued models
03E50 Continuum hypothesis and Martin’s axiom
54D15 Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.)
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