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I am reading about projective limits, and I am getting confused. In this answer, it is stated that the profinite spaces are either finite or uncountable. However, in the following example, I am constructing a countably infinite profinite space. I do not understand where I am making a mistake. The following example I constructed after reading an answer here.

Suppose $S=\mathbb{N}$, and for each $n \in \mathbb{N}$, $\Pi_n=\{[1], [2],\ldots, [n-1], [n, n+1, \ldots]\}$ is a partition of $S$, and for each $n\geq m$, there is a canonical projection $\pi_{mn}:\Pi_n\to\Pi_m$ that maps each cell in $\Pi_n$ to the unique cell in $\Pi_m$ that contains the former as a subset. Then $(\Pi_n)$ forms a projective system, and its projective limit is countably infinite. Now if we consider discrete topology on the above partitions, then the projective limit, which is a profinite space, is countably infinite.

Hakuna Matata
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  • Why is the projective limit countable? – KCd Aug 02 '23 at 21:13
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    Let us denote $S_n=\mathbb{N} \setminus {1,2, \ldots, n-1}$. The elements of the projective limit are the following:
    1. for each $n \in \mathbb{N}$, the element $(\ldots, S_n, S_n, S_n, \ldots, S_3, S_2, S_1)$ is in the projective limit
    2. Besides the above elements, the remaining element is $(\ldots, S_3, S_2, S_1)$.

    Thus the projective limit is countable.

     – Hakuna Matata Aug 02 '23 at 21:24

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It's not true that profinite spaces (also called Stone spaces) are either finite or uncountable. Your example works to show this. You could also see this by considering $\{0\}\cup \{\frac{1}{n}\mid n\in \mathbb{N}_+\}$ as a subspace of $\mathbb{R}$, or the ordinal $\omega+1$ with the order topology, or the one point compactification of a countably infinite discrete space. All of these spaces are homeomorphic to the projective limit in your example, and all are easily seen to be compact, Hausdorff, and totally disconnected.

In the answer you linked to, it's possible that the answerer was thinking of the fact that profinite groups are always finite or uncountable.

Alex Kruckman
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