The minitightness of products (Q290607)

In the present paper the minitightness (a.k.a. the weak functional tightness) of the product of Tychonoff spaces is investigated. The minitightness was defined by V. Arhangelskii.NEWLINENEWLINEFor a space \(X\) and an infinite cardinal number \(k\), denote by \(\beta _{k} X\) the \(k\)-closure of \(X\) in \(\beta X\).NEWLINENEWLINEFrom propositions 1.4 and 1.7 it follows directly that the following conditions are equivalent:NEWLINENEWLINE(i) \(t_{m} X \leq k\);NEWLINENEWLINE(ii) for every space \(Y\) every strictly \(k\)-continuous function from \(X\) to \(Y\) is continuous;NEWLINENEWLINE(iii) for every compact space \(Y\) every strictly \(k\)-continuous function from \(X\) to \(Y\) is continuous;NEWLINENEWLINE(iv) every strictly \(k\)-continuous function from \(X\) to \([0,1]\) is continuous;NEWLINENEWLINE(v) \(t_{m} \beta _{k} X \leq k\).NEWLINENEWLINEIn this paper the following theorems are proved:NEWLINENEWLINETheorem 2.7. For any spaces \(X\) and \(Y\) NEWLINE\[NEWLINE t_{m}(X \times Y) \leq t_{m}(X) \chi (Y)NEWLINE\]NEWLINENEWLINENEWLINETheorem 2.14. If \(X\) is a locally compact space, then for every space \(Y\), \(t_{m} (X \times Y) \leq t_{m}(X) t_{m} (Y)\).NEWLINENEWLINETheorem 2.13. Let \(U\) be an open subset of \(X\). Then \(t_{m} (U) \leq t_{m} (X)\).