Space Suggestion: Open extension = Non-Hausdorff cone of Circle, equivalently, Leaf space of Reeb component S^1 x D^n, n > 1

[GeoffreySangston]

Space Suggestion

Let $X = S^1 \sqcup \{p\}$. A nonempty set is open if and only if it equals $X$ or is an open subset of $S^1$. This is the open extension of Circle. The open extension construction is described in general in item #8 for space #16 of Steen-Seebach. This also equals the non-Hausdorff cone over $S^1$. The non-Hausdorff cone is described in general in Definition 1.8.2 of Peter May's Finite Spaces and Larger Contexts. Based on the convention established in S50, the main name should probably be Circle extended by a focal point.

$X$ is homeomorphic to the leaf space of a Reeb component $S^1 \times D^n$, $n \ge 2$. See Example 1 of Rosenberg and Roussarie's Reeb foliations for the case $n = 2$. Thurston in Exercise 3.1.11 and Figure 3.1 of Three-Dimensional Geometry and Topology shares the following algebraic model for a Reeb component. Foliate $\mathbb{R}^n \backslash \{0\}$ by planes (one is punctured) parallel to $\mathbb{R}^{n-1} \times \{0\}$. Since this foliation is preserved by the homothety $\lambda : \mathbb{R}^n \to \mathbb{R}^n$, $x \mapsto 2x$, it passes to a foliation of the quotient $\langle \lambda \rangle \backslash (\mathbb{R}^n \backslash \{0\}) \approx S^1 \times S^{n-1}$. The subspace $\langle \lambda \rangle \backslash ((\mathbb{R}^{n-1} \times \{0\}) \backslash \{0\}) \approx S^1 \times S^{n-2}$ is the boundary of two Reeb components, given by the foliation restricted to $\langle \lambda \rangle \backslash ((\mathbb{R}^{n-1} \times \{c\}) \backslash \{0\})$ such that $c \ge 0$, or such that $c \le 0$.

• Lawson's paper on Project Euclid looks like a nice introductory reference to foliations which is available online.

Sort of mentioned in #1158, $X$ is homeomorphic to the orbit space of the action of $\langle 2 \rangle \subset \mathbb{R}^\ast$ on $[0, \infty)$. The orbit space on $(0, \infty)$ is called a Hopf circle in the context of $(G, X)$ manifolds (not to be confused with Hopf fibration). (Aside, Hopf circles are interesting because they're the models for incomplete geodesics in flat affine manifolds, which is interesting because compactness implies geoesic completeness for (Riemannian) Levi-Civita connections. There's a family of these parameterized by $\mathbb{R}_+$, where the parameter is sort of of like a radius, except it corresponds to the magnitude a constant speed particle has accelerated relative to fixed objects after a revolution around the Hopf circle.)

Rationale

The real reason I'm interested in this is of course due to the Reeb component representatives, which are important examples of codimension one foliations. The leaf spaces provide naturally occurring examples of spaces with a focal point, which are different in character than the examples already in pi-base, and which illustrate that concept in an interesting way. I think it would be helpful to have this space here even if it wasn't uniquely characterized in pi-base, however. I'm in the beginner stage with this subject so feel free to push back on anything.

Relationship to other spaces and properties

It would be the first example of either of the following searches

• π-Base, Search for Has a point with a unique neighborhood + ~Countable + ~indiscrete + ~almost discrete
• π-Base, Search for Has a point with a unique neighborhood + Has a closed point + second countable + Cardinality $=\mathfrak c$

The following only has two examples, so the other conditions are about eliminating those.

• π-Base, Search for Has a point with a unique neighborhood + ~Countable

Other remarks

Not suggesting this here, but for future reference: I think the leaf space of the annuluar Reeb component $S^1 \times [-1, 1]$ is the non-Hausdorff suspension of $S^1$ from Peter May, and it seems fortuitous that the non-Hausdorff suspension of $S^1$ is also what's used in Peter May to illustrate the concept (without mentioning Reeb components however). I think this space satisfies
π-Base, Search for ~hausdorff + simply connected + ~contractible + Has a closed point.

• Peter May's Lemma 3.4.9 gives a weak homotopy equivalence from the standard suspension to the non-Hausdorff suspension.

The leaf space of the famous Reeb foliation of $S^3$ is the non-Hausdorff cone over $S^1 \sqcup S^1$. For this one the focal point is a cut point, and $X$ has no cut points, so this one would be uniquely characterized in pi-base as well.