multisequence product of sequential spaces
On multisequences and their application to products of sequential spaces
Let \(X\) be a topological space and let \(\operatorname {cl}\) be the closure operator in \(X\). A sequential adherence of \(A\subset X\) is the set of all limits of convergent sequences ranging in \(A\). The corresponding sequential adherence operator \(\operatorname {cl}_{\text{seq}}\) need not be idempotent and can be iterated up to \(\omega_1\) times -- this way we get an idempotent operator \(\operatorname {cl}_{T\operatorname {seq}}\) called the sequential closure of \(A\). If \(\operatorname {cl}_{\operatorname {seq}}=\operatorname {cl}\), then \(X\) is a Fréchet space and if \(\operatorname {cl}_ {T\operatorname {seq}}=\operatorname {cl}\), then \(X\) is a sequential space. The sequential order \(\sigma (X)\) of \(X\) is at most \(\omega_1\) and it is roughly ``how many times we have to iterate \(\operatorname {cl}_{\operatorname {seq}}\) to get \(\operatorname {cl}_{T\operatorname {cl}}\)''. NEWLINENEWLINENEWLINEThe paper under review deals with sequential properties of products. The main result (Theorem 3.3): If \(X\) is a sequential space and \(Y\) is a sequential regular locally countably compact space, then \(X\times Y\) is sequential and \(\sigma (X\times Y)\leqq \sigma (X) + \sigma (Y)\). The result generalizes several known theorems about products by A. V. Arkhangel'skij, E. Michael, T. Nogura and A. Shibakov. NEWLINENEWLINENEWLINEFrom the technical viewpoint, all is based on the convergence of multisequences -- a useful but rather technical construction. If \(x\in \operatorname {cl}_{T\operatorname {seq}} A\), then there is a sequence \((x(n))\) of points ``sequentially closer to \(A\) than \(x\)'', each \(x(n)\) is a limit of a sequence \((x(n,m))\) of points ``sequentially closer to \(A\) than \(x(n)\)'', an so on. Due to the compactness of \(\omega_1+1\) the ``fibres'' are finite and we always end up in \(A\) after finitely many steps. Hence multisequences ``live within the set of finite sequences of natural numbers'' and their convergence is defined in terms of ``maximal elements of index trees''. NEWLINENEWLINENEWLINEThe multisequence technique has been used by \textit{P. Kratochvíl} [Gen. Topol. Relat. mod. Anal. Algebra, Proc. 4th Prague Topol. Symp. 1976, Part B, 237-244 (1977; Zbl 0383.54006)] and by \textit{D.~H. Fremlin} [Commentat. Math. Univ. Carol. 35, No. 2, 371-382 (1994; Zbl 0827.54002)]. Recently, multisequences have been systematically used by \textit{S. Dolecki} and \textit{S. Sitou} to get a better understanding of sequential properties of products, see, e.g. [Topology Appl. 84, No. 1-3, 61-75 (1998; Zbl 0921.54020)].
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