In [12], John Stillwell wrote, ‘finding the exact strength of the Brouwer invariance theorems seems to me one of the most interesting open problems in reverse mathematics.’ In this article, we solve Stillwell’s problem by showing that (some forms of) the Brouwer invariance theorems are equivalent to the weak König’s lemma over the base system ${\sf RCA}_0$ . In particular, there exists an explicit algorithm which, whenever the weak König’s lemma is false, constructs a topological embedding of $\mathbb {R}^4$ into $\mathbb {R}^3$ .

中文翻译：

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逆向数学中的 Brouwer 不变定理

在 [12] 中，John Stillwell 写道，“在我看来，找到 Brouwer 不变性定理的确切强度似乎是逆向数学中最有趣的开放问题之一。” 在本文中，我们通过证明（某些形式的）Brouwer 不变性定理等价于基系统上的弱 König 引理来解决 Stillwell 问题 ${\sf RCA}_0$ . 特别是，存在一个显式算法，只要弱 König 引理为假，它就会构造一个拓扑嵌入 $\mathbb {R}^4$ 进入 $\mathbb {R}^3$ .