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 Łos theorem”). Tychonoff products, box products and ultraproducts appear as special cases of the general construction.

We compare the forcing-related properties of a complete Boolean algebra $\mathbb{B}$ with the properties of the convergences ${\lambda}_{\mathrm{s}}$ (the algebraic convergence) and ${\lambda}_{\mathrm{ls}}$ on $\mathbb{B}$ generalizing the convergence on the Cantor and Aleksandrov cube, respectively. In particular, we show that ${\lambda}_{\mathrm{ls}}$ is a topological convergence iff forcing by $\mathbb{B}$ does not produce new reals and that ${\lambda}_{\mathrm{ls}}$ is weakly topological if $\mathbb{B}$ satisfies condition $\left(\hslash \right)$ (implied by the $\U0001d531$-cc). On the other hand, if ${\lambda}_{\mathrm{ls}}$ is a weakly topological convergence, then $\mathbb{B}$ is a ${2}^{\U0001d525}$-cc algebra...

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