# An interesting topological space with $4$ elements

There is an interesting topological space $X$ with just four elements $\eta,\eta',x,x'$ whose nontrivial open subsets are $\{\eta\},\{\eta'\},\{\eta,\eta'\}, \{\eta,x,\eta'\}, \{\eta,x',\eta'\}$. This seems to be some "discrete model" for the $1$-sphere $S^1 \subseteq \mathbb{C}$: The open sets $\{\eta,x,\eta'\}$ and $\{\eta,x',\eta'\}$ may be imagined as arcs joining $\eta$ and $\eta'$ via $x$ or resp. $x'$. They are contractible, and their intersection is the discrete space $\{\eta\} \sqcup \{\eta'\}$. It also follows that $\pi_1(X) \cong \mathbb{Z}$. Without knowing any algebraic topology, one can explicitly classify all coverings of $X$, namely this category is equialent to $\mathbb{Z}\mathsf{-Sets}$.

In contrast to $S^1$, actually $X$ is homeomorphic to the spectrum of a ring: Let $R$ be the localization of $\mathbb{Z}$ at all primes $p \neq 2,3$. There is a canonical surjective homomorphism $R \to \mathbb{Z}/6$. Let $A$ be the fiber product $R \times_{\mathbb{Z}/6} R$. Then $X \cong |\mathrm{Spec}(A)|$. We glue $\mathrm{Spec}(R) = \{\eta,x,x'\}$ with itsself along its closed subscheme $\{x,x'\}$. We end up with two generic points $\eta,\eta'$.

Question. Does this space $X$ have a name? What is the precise relationship to $S^1$? Where do the observations above appear in the literature?

According to Miha's comment below (which is an answer), $X$ is called the pseudocircle, and the definite source for the general phenomenon is:

Singular homology groups and homotopy groups of finite topological spaces, by Michael C. McCord, Duke Math. J., 33(1966), 465-474, doi:10.1215/S0012-7094-66-03352-7.

The pseudocircle already appeared a couple of times on math.SE, for example in question/56500.

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This seems to be the pseudocircle. –  Miha Habič Jun 22 '12 at 9:54
@Miha: This is an answer, not a comment. –  Martin Brandenburg Jun 22 '12 at 9:57
I was really sort of hoping someone would come along eventually with some more information than just a wiki link. –  Miha Habič Jun 22 '12 at 9:59
The name already gives me a lot of information where to look - I would accept that as an answer. –  Martin Brandenburg Jun 22 '12 at 10:16
Also see my answer to the question here: math.stackexchange.com/questions/80217/… –  wckronholm Jun 22 '12 at 17:05

According to Miha's comment below (which is an answer), $X$ is called the pseudocircle, and the definite source for the general phenomenon is: