Topologist's sine curve

The topologist's sine curve is defined as the following subspace of Real coördinate space $ℝ 2$ #m/def/topology

𝐿 = {(0, 𝑦) : 𝑦 \in [- 1, 1]} 𝑆 = {(𝑥, s i n (1 / 𝑥)) : 𝑥 \in (0, 1]} 𝑋 = 𝐿 \cup 𝑆

Specifically, defined above is the compact variant.

Properties

$𝑆$ is connected, but not path-connected.

Proof

Let $𝜄 : 𝑋 ↪ ℝ 2$ denote the inclusion and $𝜋 𝑥 : ℝ 2 ↠ ℝ$ denote the projection onto the $𝑥$ -axis, so $𝜋 𝑥 𝜄 : 𝑋 \to ℝ$ is the continuous projection of $𝑋$ onto the $𝑥$ -axis. Suppose $𝑝 : 𝕀 \to 𝑋$ is a continuous path from $(1, 0)$ to $(0, 0)$ . It follows by the Intermediate value theorem that the image $𝜋 𝑥 𝜄 𝑓 (𝕀) = 𝕀$ . Let $𝜏 = s u p {𝑡 : 𝑝 (𝑡) \in 𝑆}$ . Clearly $𝑝 (𝜏) \notin 𝑆$ , for if it were we could find a $˜ 𝜏 > 𝜏$ such that $𝜋 𝑥 𝜄 𝑝 (˜ 𝜏) \in (0, 𝑝 (𝜏))$ by the intermediate value theorem. Again invoking the intermediate value theorem, there exists an increasing sequence $(𝑡 𝑛) \infty 𝑛 = 1$ such that $𝜋 𝑥 𝜄 𝑝 (𝑡 𝑛) = 2 𝜋 (2 𝑛 + 1)$ . Now $(𝑡 𝑛) \to 𝜏$ (why?), so by sequential continuity $𝑝 (𝑡 𝑛) \to 𝑝 (𝜏)$ . But $𝑝 (𝑡 𝑛)$ is not convergent, since its $𝑦$ -coördinate alternates between $- 1$ and $1$ , a contradiction. Therefore $𝑝$ cannot be a continuous path.

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