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separation axioms; box product; ultraproduct; ideal product topology
We give sufficient and necessary conditions to be fulfilled by a filter $\Psi $ and an ideal $\Lambda $ in order that the $\Psi $-quotient space of the $\Lambda $-ideal product space preserves $T_k$-properties ($k=0,1,2,3,3\frac{1}{2}$) (``in the sense of the \L os theorem''). Tychonoff products, box products and ultraproducts appear as special cases of the general construction.
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