Statistical convergence and porosity of sets (Q2774490)

The authors deal with the porosity of sets \(C^0_{\operatorname {stat}}\) (the set of all bounded statistically convergent sequences) in the space \(m\) (of all bounded real sequences with the sup-norm) and \(C_{\operatorname {stat}}\) (the set of all statistically convergent sequences) in the space \(s\) (of all real sequences endowed with Fréchet's metric). The notion of \(h\)-porosity in a metric space \((Y,\rho)\) is introduced in the following way: Let \(h\:[0, +\infty)\to \mathbb R\) be a non-negative continuous function increasing on \([0, +\infty)\) with \(h(0) = 0\), \(h(x)>0\) for \(x>0\). Let \(B(y,\delta) = \{x\in Y\mid \rho (x,y)<\delta \}\), \(\gamma (y,\delta , M) = \sup\{ t>0\mid\) there exists \(z\in B(y,\delta)\) such that \(B(z,t)\subset B(y,\delta)\) and \(B(z,t)\cap M =\emptyset \}\), where \(M\subset Y\), \(y\in Y\), \(\delta > 0\). Let \(\gamma _h(y,\delta , M) =h(\gamma (y,\delta ,M))\). The number \(\underline p_h(y,M)= \liminf_{\delta \to 0+}\gamma _h (y,\delta ,M)/h(\delta)\), \((\overline p_h(y,M)=\limsup_{\delta \to 0+}\gamma _h(y,\delta ,M)/h(\delta))\) is said to be the lower (upper) \(h\)-porosity of \(M\) at \(y\). If \(\underline p_h(y,M) =\overline p_h(y,M)\) \((=p_h(y,M))\), then \(p_h(y,M)\) is said to be the \(h\)-porosity of \(M\) at \(y\). A set \(M\subset Y\) is said to be very \(c\)-\(h\)-porous at \(y\) provided that \(p_h(y,M) \geq c >0\). A set \(M\subset Y\) is said to be uniformly \(\sigma \)-very \(c\)-\(h\)-porous in \(Y\) if \(M=\bigcup _{n=1}^{\infty } M_n\) and each \(M_n\) is very \(c\)-\(h\)-porous at each \(y\in Y\). If \(h(x)=x\), then \(h\)-porosity coincides with usual porosity and \(p_h=p\). The main results of the paper follow. NEWLINENEWLINENEWLINETheorem. (1) If \(y\in m-C^0_{\operatorname {stat}}\), then \(p(y, C^0_{\operatorname {stat}}) = 1.\) (2) If \(y\in C^0_{\operatorname {stat}}\), then \(p(y,C^0_{\operatorname {stat}}) = \frac 12\). NEWLINENEWLINENEWLINETheorem. Let \(0<\varepsilon <1\). Then the set \(C_{\operatorname {stat}}\) is uniformly \(\sigma \)-very \((1 - \varepsilon)\)-\(h\)-porous in \(s\), where \(h(0) =0\), \(h(x) = |\ln x|^{-1}\) for \(0<x<e^{-1}\), \(h(x) = x - e^{-1} +1\) for \(x\geq e^{-1}\). NEWLINENEWLINENEWLINEThe paper improves some results of the second author [On statistically convergent sequences of real numbers, Math. Slovaca 30, 139-150 (1980; Zbl 0437.40003)].