The Collatz conjecture establishes that for every natural number n ∈ ℕ, there exists an r ∈ ℕ such that κr (n) = 1, where κ : ℕ → ℕ is the function defined as n/2 if n es even and as 3n + 1 if n is odd. The map κ induces a topology τκ on ℕ. We prove that the Collatz conjecture and connectedness of the space (ℕ, τκ) imply that the space is simply connected. Furthermore, we prove that the Collatz conjecture is equivalent to the fact that the space (ℕ, τκ) is path-connected.

中文翻译：

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原始空间中的路径和 Collat​​z 猜想

Collat​​z 猜想建立了对于每个自然数 n ∈ ℕ，存在一个 r ∈ ℕ 使得 κr (n) = 1，其中 κ : ℕ → ℕ 是定义为 n/2 如果 n 是偶数和 3n + 的函数如果 n 是奇数，则为 1。映射 κ 在 ℕ 上引入了拓扑 τκ。我们证明了 Collat​​z 猜想和空间的连通性 (ℕ, τκ) 意味着空间是简单连通的。此外，我们证明了 Collat​​z 猜想等价于空间 (ℕ, τκ) 是路径连通的。