First, a bit of background on orbifolds:

Let $X$ be a connected (effective) orbifold. To every point $x \in X$, we associated a group $G_x$ called the

isotropy group. The singular locus $\Sigma X$ is the set of points $x$ for which $G_x \neq 1$, and these points are calledsingular points. Non-singular points are calledregular. The singular locus is a closed, nowhere dense subset of $X$, and the set of regular points $X^r$ is open and dense in $X$.The orbifold $X$ has a

stratificiationintostrataof points of equal type, i.e. $X$ is the disjoint union of connected components along which the isotropy group is constant (up to isomorphism). In particular, the strata of codimension $0$ form the regular points, and the strata of codimension $\geq 1$ form the singular locus.

Now, my question: **is the set of regular points $X^r$ connected?**

I've done a bit of background research, but I seem to find contradictory statements. Moreover, the original definition of orbifolds (i.e. Satake's V-manifolds) assumed that there were no codimension $1$ strata, in which case the statement is clearly true. To add to the confusion, some authors still use this definition.

So, what have I found so far:

**Suggesting "Yes":**

- "This stratum is a connected manifold ..." [
*Riemannian orbifolds with non-negative curvature*, Dmytro Yeroshkin] - "The regular points of an orbifold form the top-dimensional stratum. It is open, dense and path-connected." [
*The Topology of locally volume collapsed 3-Orbifolds*, Daniel Faessler] - "Observe that the set of regular points is a dense
*connected*open subset of the topological space underlying a connected orbifold" [*Seifert Fibred 3-Orbifolds*, Bonahon & Siebenmann] - "Observe that the set of regular points is a dense connected open subset of $X$" [
*Lusternik-Schnirelmann category of Orbifolds*, Hellen Colman] - "The regular part of an orbifold is connected." (
**proof given, which uses that the regular part is locally connected**) [*Orbifolds from a metric viewpoint*, Christian Lange]

**Suggesting "No":**

- "Having done this, the set of special points has codimension at least 2. The set of regular points is therefore a connected manifold." [
*Differential Topology, Foliations, and Group Actions*, Paul A. Schweitzer] - "...we require that
the fixed-point set is of codimension at least two. [...] This requirement has the consequence that the non-fixed-point set is locally connected." [
*Stringy geometry and topology of orbifolds*, Yongbin Ruan]

saysyes. $\endgroup$ – Lee Mosher Apr 12 '18 at 12:13