Verbal covering properties of topological spaces

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• 作者：Taras Banakh^a ; ^b ; ^{t.o.banakh@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; Alex Ravsky^c ; ^{oravsky@mail.ru" class="auth_mail" title="E-mail the corresponding author}
• 关键词：54A25 ; 54D20
• 刊名：Topology and its Applications
• 出版年：2016
• 出版时间：15 March 2016
• 年：2016
• 卷：201
• 期：Complete
• 页码：181-205
• 全文大小：602 K

文摘

For any topological space X   we study the relation between the universal uniformity U_X${\mathcal{U}}_X$, the universal quasi-uniformity 52bb97916c3a4e67" title="Click to view the MathML source">qU_X$q{\mathcal{U}}_X$ and the universal pre-uniformity pU_X$p{\mathcal{U}}_X$ on X  . For a pre-uniformity U$\mathcal{U}$ on a set X and a word v   in the two-letter alphabet {+,−}$\{+,-\}$ we define the verbal power U^v${\mathcal{U}}^v$ of U$\mathcal{U}$ and study its boundedness numbers ℓ(U^v)$\ell ({\mathcal{U}}^v)$, $\overline{\ell }({\mathcal{U}}^v)$, L(U^v)$L({\mathcal{U}}^v)$ and $\overline{L}({\mathcal{U}}^v)$. The boundedness numbers of (the Boolean operations over) the verbal powers of the canonical pre-uniformities pU_X$p{\mathcal{U}}_X$, 52bb97916c3a4e67" title="Click to view the MathML source">qU_X$q{\mathcal{U}}_X$ and U_X${\mathcal{U}}_X$ yield new cardinal characteristics ℓ^v(X)${\ell }^v(X)$, ${\overline{\ell }}^v(X)$, L^v(X)$L^v(X)$, ${\overline{L}}^v(X)$, 52bd835bbe80b4a7a3a51eaaf151cc" title="Click to view the MathML source">qℓ^v(X)$q{\ell }^v(X)$, $q{\overline{\ell }}^v(X)$, qL^v(X)$q{L}^v(X)$, $q{\overline{L}}^v(X)$, uℓ(X)$u\ell (X)$ of a topological space X  , which generalize all known cardinal topological invariants related to (star-)covering properties. We study the relation of the new cardinal invariants b99bc3731066186352f6f4477e9f" title="Click to view the MathML source">ℓ^v${\ell }^v$, ${\overline{\ell }}^v$ to classical cardinal topological invariants such as Lindelöf number ℓ, density d, and spread s  . The simplest new verbal cardinal invariant is the foredensity ℓ⁻(X)${\ell }^{-}(X)$ defined for a topological space X as the smallest cardinal κ   such that for any neighborhood assignment (O_x)_x∈X${(O_x)}_{x\in X}$ there is a subset A⊂X$A\subset X$ of cardinality |A|≤κ$|A|\le \kappa$ that meets each neighborhood O_x$O_x$, x∈X$x\in X$. It is clear that ℓ⁻(X)≤d(X)≤ℓ⁻(X)⋅χ(X)${\ell }^{-}(X)\le d(X)\le {\ell }^{-}(X)\cdot \chi (X)$. We shall prove that ℓ⁻(X)=d(X)${\ell }^{-}(X)=d(X)$ if |X|<ℵ_ω$|X|<{\mathrm{\aleph }}_{\omega }$. On the other hand, for every singular cardinal κ   (with κ≤2^2cf(κ)$\kappa \le 2^{2^{\mathrm{cf}(\kappa )}}$) we construct a (totally disconnected) T₁$T_1$-space X   such that ℓ⁻(X)=cf(κ)<κ=|X|=d(X)${\ell }^{-}(X)=\mathrm{cf}(\kappa )<\kappa =|X|=d(X)$.