Our main objective is a further study of $\mathcal{M}$-factorizability in topological groups as defined in Zhang, Peng, He, Tkachenko (2020) [15]. We focus on topological-algebraic implications of $\mathcal{M}$-factorizability such as τ-precompactness, pseudo-τ-compactness and τ-fineness. We also study products of topological groups and present necessary and sufficient conditions on the factors guaranteeing the $\mathcal{M}$-factorizability of products. Our main technical tool for this study is the new notion of τ-fine topological group, where $\tau >\omega$ is a cardinal. We prove the following dichotomy theorem: Every $\mathcal{M}$-factorizable topological group is either $\mathbb{R}$-factorizable or ${\omega }_1$-fine.

Another dichotomy is established for the product of two groups. We prove that if the product $G\times H$ of topological groups is $\mathcal{M}$-factorizable, then for every cardinal $\tau >\omega$, either G is τ-fine or H is pseudo-τ-compact. We also show that the product $G\times H$ is $\mathcal{M}$-factorizable provided G is a metrizable topological group with $w(G)\le \tau$ and H is a τ-fine topological group with $hl(H)\le \tau$.

It is also proved that the product $G\times H$ is $\mathcal{M}$-factorizable ($\mathbb{R}$-factorizable) whenever G is an arbitrary $\mathcal{M}$-factorizable ($\mathbb{R}$-factorizable) topological group and H is a locally compact separable metrizable topological group.

我们的主要目标是对 $\mathcal{中号}$Zhang，Peng，He，Tkachenko（2020）中定义的拓扑群中的可分解性[15]。我们专注于...的拓扑-代数含义$\mathcal{中号}$-可分解性，例如τ-预紧实度，伪τ-紧实度和τ细度。我们还研究拓扑组的乘积，并就保证安全性的因素提出了充分必要的条件。$\mathcal{中号}$产品的可分解性。这项研究的主要技术工具是τ-精细拓扑群的新概念，其中$\tau >\omega$是红衣主教 我们证明以下二分定理：$\mathcal{中号}$可分解的拓扑组是 $\mathbb{[R}$可分解或 ${\omega }_{\mathrm{1个}}$-美好的。

建立了针对两组产品的另一种二分法。我们证明如果产品$G\times H$ 的拓扑组是 $\mathcal{中号}$-可分解的，然后针对每个基数 $\tau >\omega$，G是τ- fine或H是伪τ- compact。我们还展示了该产品$G\times H$ 是 $\mathcal{中号}$可分解的，前提是G是具有以下特征的可化拓扑组$w（G）\le \tau$和ħ是τ -精细拓扑群与$H升（H）\le \tau$。

也证明了该产品 $G\times H$ 是 $\mathcal{中号}$可分解的（$\mathbb{[R}$-factorizable）只要G是任意的$\mathcal{中号}$可分解的（$\mathbb{[R}$-可分解的拓扑组，而H是局部紧凑的可分离的可量化的拓扑组。