Cantor的漏帳篷

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这里举了单连通但局部路径不连通的拓扑空间的例子:Knaster–Kuratowski fan / Cantor's leaky tent
设 $C$ 为康托集，设 $p$ 为点 $\left({\frac {1}{2}},{\frac {1}{2}}\right)\in ℝ ^2$ .
对于任何 $c\in C$，用 $L(c)$ 表示连接 $(c,0)$ 和 $p$ 的线段。
如果 $c\in C$ 是在 Cantor 集中删除的区间的端点，则设 $X_{{c}}=\{(x,y)\in L(c):y\in {\mathbb {Q}}\}$；
对于 $C$ 中的所有其他点，设 $X_{{c}}=\{(x,y)\in L(c):y\notin {\mathbb {Q}}\}$;
Knaster–Kuratowski 扇定义为 $\bigcup _{{c\in C}}X_{{c}}$，配备从 $\mathbb {R} ^{2}$ 上的标准拓扑继承的子空间拓扑.
The fan itself is connected, but becomes totally disconnected upon the removal of $p$.
Cantor’s leaky tent is connected but just barely. The top point p is like a string hanging off a sweater: if you tug on it, the whole thing unravels. When we remove p, not only does the set become disconnected, it becomes as disconnected as possible. You can separate it into two disjoint pieces fairly easily: one of the pieces contains points with x coordinates less than 1/2 and one with x coordinates greater than 1/2. But it’s even messier: no two points in the new set, sometimes called Cantor’s teepee, are in the same connected piece. For any two points, you can always find a way to separate Cantor’s teepee into subsets that keep the points apart. Cantor’s Leaky Tent has just one piece, but Cantor’s teepee has infinitely many, and they’re just single points. If you’re interested in the lingo, p is called a dispersion point, and sets that don't have any connected pieces bigger than a point are called totally disconnected.

π-Base Cantor's leaky tent is not recommended for use in inclement weather. 證明Cantor的漏帳篷是連通的

Zero-dimensionality of Cantor's leakier tent
How is the Knaster–Kuratowski fan connected?