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I have been doing some reading on general topology, connectedness in particular. Here is a question on a topological concept called quasi-component. Here is a definition: Let $I = [0,1]$. Consider the subset of the real numbers:

$$S = \left \{\frac{1}{n} \in \mathbb{R} \ | \ n \in \mathbb{N} \right \}$$

Consider the space: $$X = (S \times I) \cup \{(0,0), (0,1) \}$$

What are the components and the path-components and the quasi-components of $X$? For a definition of path-components, please see here:

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up vote 3 down vote accepted

The path components are the sets $\left\{\frac1n\right\}\times I$ for $n\in\Bbb Z^+$ and the two singletons $\{\langle 0,0\rangle\}$ and $\{\langle 0,1\rangle\}$. The only part of that that isn’t completely obvious is that there is no path between $\langle 0,0\rangle$ and $\langle 0,1\rangle$, and the first proof given here for the same space but including $I\times\{0\}$ goes through for your $X$.

It’s also clear that the sets $\left\{\frac1n\right\}\times I$ for $n\in\Bbb Z^+$ are both components and quasi-components: each is a maximal connected set in $X$, and each is clopen in $X$. The singletons $\{\langle 0,0\rangle\}$ and $\{\langle 0,1\rangle\}$ are also components. They are not quasi-components, however: you should try to prove that any clopen set in $X$ that contains one of those points also contains the other. HINT: If $H$ is a clopen set in $X$ such that $$H\cap\left(\left\{\frac1n\right\}\times I\right)\ne\varnothing\;,$$ then $$H\supseteq\left\{\frac1n\right\}\times I\;.$$

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