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This is a curious example I'm trying to understand.

For any positive integer $n$, let $A_n=\left\{\dfrac{1}{n}\right\}\times[0,1]$, and let $X=\bigcup_{n}A_n\cup\{(0,0)\}\cup\{(0,1)\}$. Viewing $X$ as a subset of the Euclidean plane, we can equip it with the relative topology.

Then why are $\{(0,0)\}$ and $\{(0,1)\}$ are components of $X$ and each closed and open subset contains both or neither of the points, but never just one or the other?

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This is very similar to what is called comb space: check this out – Bruno Stonek Dec 11 '11 at 19:28
up vote 2 down vote accepted

There are two things to address here:

1) Why are the singletons $\{(0,1)\}$ and $\{(0,0)\}$ components?

2) Why is there no closed and open subset of $X$ containing one point and not the other?

For the first question, recall that a component of $X$ is defined as a maximal connected subspace. So we need to show that the only connected subspace of $X$ containing, say, $(0,0)$, is the singleton set. That is, we need to show that a connected subspace $C \subset X$ containing $(0,0)$ must be the singleton. It should be readily apparent that $C$ cannot contain any point of $A_n$ for any $n$, because those points can be separated by a vertical line with irrational $x$-coordinate, and it immediately follows that it can't contain $(0,1)$ either.

As for the second question, any closed and open subset of $X$ must contain either all of or none of $A_n$, for each $n \in \mathbb{N}$. This is because $A_n$ is connected. So if $C \subset X$ is both open and closed, and contains $(0,0)$, then it contains $A_n$ for $n$ sufficiently large, ie, close to $(0,0)$ and $(0,1)$. Therefore $(0,1)$ is a limit point of $C$.

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