In this paper, we investigate the existence of predecessors and successors of the usual Euclidean topology ${\tau }_X$ on X in $\mathcal{PG}(X)$ and $\mathcal{G}(X)$, where $X=\{\left(\begin{array}{cc}\hfill a\hfill & \hfill b\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right):a>0,a,b\in \mathbb{R}\}$ and $\mathcal{G}(X)$ ($\mathcal{PG}(X)$) is the lattice of all topological (paratopological) group topologies on X. We give a complete description of predecessors of ${\tau }_X$ in $\mathcal{G}(X)$ based on the fact that $(X,{\tau }_X)$ is a minimal Hausdorff topological group which was shown by Dierolf and Schwanengel in 1979. Then we give a negative answer to an open problem posed in [8]. Some constructions of successors of ${\tau }_X$ in $\mathcal{PG}(X)$ are also given. We also prove that ${\tau }_X$ has no successors in $\mathcal{G}(X)$.

在本文中，我们研究了通常的欧几里得拓扑的前辈和后继者的存在 ${\tau }_X$在X上$\mathcal{PG}（X）$ 和 $\mathcal{G}（X）$， 在哪里 $X=\{（\begin{array}{cc}\hfill \mathrm{一个}\hfill & \hfill b\hfill \\ \hfill 0\hfill & \hfill \mathrm{1个}\hfill \end{array}）：\mathrm{一个}>0，\mathrm{一个}，b\in \mathbb{[R}\}$ 和 $\mathcal{G}（X）$ （$\mathcal{PG}（X）$）是X上所有拓扑（超拓扑）组拓扑的晶格。我们对以下产品的前身进行了完整的描述${\tau }_X$ 在 $\mathcal{G}（X）$ 基于以下事实 $（X，{\tau }_X）$是一个最小的Hausdorff拓扑群，由Dierolf和Schwanengel在1979年展示。然后我们对[8]中提出的一个开放问题给出否定答案。的后续结构${\tau }_X$ 在 $\mathcal{PG}（X）$还给出了。我们还证明${\tau }_X$ 在没有继任者 $\mathcal{G}（X）$。