Nested homeomorphic sets

Let $$U_i = \{e^{2\pi i \theta} -1 \mid \theta \in [0,1-1/i)\} \cup \{e^{2\pi i \theta} +1 \mid \theta \in [1/2, 3/2-1/i)\}.$$

Then every $U_i$ is a subset of the union of two circles that are glued together at a single point at the origin and is homeomorphic to $\mathbb{R}$. The union $\bigcup_i \geq 1$ is equal to the union of the two circles.

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Here's a simpler example for $X\cong [0,1)$ using the same idea. Let $$U_i = \{e^{2\pi i\theta} \mid \theta \in [0,1-1/i)\}.$$

Then $U_i \cong [0,1)$ for all $i \geq 1$, but $\bigcup_{i \geq 1} U_i = S^1$.