道路连通可求长吗？

TaiFanLeAi

在证明拓扑学曲线连通但不道路连通时，明显从Y轴到振荡曲线上的任何一条道路长度都无限，如果道路都可求长，，则直接就证明了

hbghlyj

EoM
A rectifiable curve is a curve having finite length (cf. Line (curve)). More precisely, consider a metric space $(X,d)$ and a continuous function $γ:[0,1]→X$. $γ$ is a parametrization of a rectifiable curve if there is an homeomorphism $φ:[0,1]→[0,1]$ such that the map $γ∘φ$ is Lipschitz.

Theorem 1 A set $E⊂\mathbb R^n$ is the image of a rectifiable curve if and only if it is compact, connected and it has finite Hausdorff 1-dimensional measure $\mathcal H^1$.

hbghlyj

MSE
This is the image of the parametric curve $γ(t)=⟨(1+1/t)\cos t,(1+1/t)\sin t⟩$, along with the unit circle. There is no path joining a point on the curve with a point on the circle. Yet, the space is the closure of the connected set $γ((1,∞))$, so it is connected.

TaiFanLeAi

搜了一下，原来道路这种映射的曲线也可以是无限长度的，比如雪花曲线。那么这种思路是不可行的。

hbghlyj

Connected planar compact set with finite length is path connected

TaiFanLeAi

额，这些我都去看了，但没太看明白。那我的这个问题，到底能不能这样证明呢？