ScalingStacks

A manifold is defined to be a topological manifold with boundary with finitely many connected components, all of which have the same dimension. A $1$-dimensional manifold is a finite disjoint union of copies of $S^{1}$, ${\mathbf{R}}$, ${\mathbf{R}}_{\geq 0}$ and $[0,1]$.

Given a point $x$ of a topological space $X$, we put $C(x)=C_{X}(x)=\lim_{U}\pi_{0}(U-\{x\})$, where $U$ runs over the set of open neighbourhoods of $x$. If $X^{\prime}$ is a subspace of $X$ containing an open neighbourhood of $x$, then we have a canonical bijection $C_{X^{\prime}}(x)\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}C_{X}(x)$ and we identify those two sets.

We put $T(X)=\coprod_{x\in X}C(x)$ and we denote by $\operatorname{pt}\nolimits:T(X)\to X$ the canonical map.

Given $E$ a finite subset of $S^{1}=\{z\in{\mathbf{C}}\ |\ ||z||=1\}$, we put $\operatorname{St}\nolimits(E)=\bigcup_{e\in E}{\mathbf{R}}_{\geq 0}e$ and $\operatorname{St}\nolimits^{\circ}(E)=\operatorname{St}\nolimits(E)-\{0\}$. These are $1$-dimensional spaces. Given $n\geq 1$, we put $\operatorname{St}\nolimits(n)=\operatorname{St}\nolimits(\{e^{2i\pi r/n}\}_{0\leq r<n})$.

Let $X$ be a $1$-dimensional space. There is a finite subset $E$ of $X$ such that $X-E$ is homeomorphic to a finite disjoint union of copies of ${\mathbf{R}}$.

Let $x\in X$. If $U$ is a small enough connected open neighbourhood of $x$, then there is a homeomorphism $U\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\operatorname{St}\nolimits(n_{x}),\ x\mapsto 0$ for some $n_{x}=n_{x,X}\geq 1$. In addition, we have a canonical bijection $C(x)\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\pi_{0}(U-\{x\})$ and we identify those two sets of cardinality $n_{x}$.

We define the boundary $\partial X=\{x\in X\ |\ n_{x}=1\}$. We put $X_{exc}=\{x\in X|n_{x}\geq 3\}$.

A $1$-dimensional space is non-singular if and only if it is a $1$-dimensional manifold.

Note that every point of a $1$-dimensional space admits a small open neighbourhood.