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On pairwise stratifiable spaces. (Spanish. English summary) Zbl 0615.54024

Let us introduce some terminology. Let \({\mathcal F}\) be a family of subsets of a topological space \((T,\tau)\). We say that: (a) \({\mathcal F}\) is a \(\sigma\)-closure preserving family if \({\mathcal F}\) is the union of countably many closure preserving subfamilies; (b) \({\mathcal F}\) is a quasi-base for \(\tau\) if, whenever \(t\in T\) and \(U\) is a neighbourhood of \(T\), there exists a set \(F\) in \({\mathcal F}\) such that \(t\in \dot F\subseteq F\subseteq U\). Now let \(\tau_ 1\) and \(\tau_ 2\) be two topologies on \(X\) and consider the bitopological space \((X,\tau_ 1,\tau_ 2)\). We say that: (c) \((X,\tau_ 1,\tau_ 2)\) is a pairwise \(M_ 2\)-space if \((X,\tau_ 1,\tau_ 2)\) is pairwise regular and, for \(i=1,2\), \(\tau_ i\) has a \(\tau_{3-i}\)-\(\sigma\)-closure preserving quasi-base; (d) \((X,\tau_ 1,\tau_ 2)\) is pairwise stratifiable space if, for every \(i=1,2\) and every \(U\in \tau_ i\), there exists a sequence \(\{U_{n,i}\}^{\infty}_{n=0}\) in \(\tau_ i\) such that \(\cup^{\infty}_{n=0}U_{n,i}=U\), \(\bar U{}_{n,i}^{\tau_{3- i}}\subseteq U\) and \(U_{n,i}\subseteq V_{n,i}\) whenever \(V\in \tau_ i\) and \(U\subseteq V.\)
The author generalizes the constructions of C. R. Borges [Pac. J. Math. 17, 1-16 (1966; Zbl 0175.198)] and J. G. Ceder [ibid. 11, 105-125 (1961; Zbl 0103.391)] and the following main results are proved: Theorem 1. Let \((X,\tau_ 1,\tau_ 2)\) be a bitopological space. If \(X\) is the union of a disjoint sequence \(\{X_ n\}^{\infty}_{n=0}\) in \(\tau_ 1\cap \tau_ 2\) such that each \((X_ n,\tau_ 1,\tau_ 2)\) is a pairwise \(M_ 2\)-space, then \(X\) is a pairwise \(M_ 2\)-space. Theorem 2. Let \((X,\tau_ 1,\tau_ 2)\) be a bitopological space. Then \((X,\tau_ 1,\tau_ 2)\) is a pairwise stratifiable space if and only if for each \(i=1,2\) and each \(U\in \tau_ i\) there exists a \(\tau_ i\)-lower semicontinuous and \(\tau_{3-i}\)-upper semi-continuous function \(f_ U: X\to [0,1]\) such that \(f_ U^{-1}(0)=X\setminus U\) and \(f_ U\leq f_ V\) whenever \(V\in \tau_ i\) and \(U\subseteq V\). Theorem 3. Let \((X,\tau_ 1,\tau_ 2)\) and \((X',\tau'_ 1,\tau'_ 2)\) be two bitopological spaces, and let \(f: X\to X'\) be a surjection. If \((X,\tau_ 1,\tau_ 2)\) is a pairwise stratifiable space and, for each \(i=1,2\), \(f: (X,\tau_ i)\to (X',\tau'_ i)\) is a closed continuous function, then \((X',\tau'_ 1,\tau_ 2)\) is a pairwise stratifiable space.
Reviewer: P.Morales

MSC:

54E55 Bitopologies
54E20 Stratifiable spaces, cosmic spaces, etc.
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