Semicircle Continuum

Semicircle Continuum is a nice continuum.

Let $A = \{(x,0)\in \Bbb R^2: 0 \le x \le 1\}$;

for each $n=1,2,\cdots$ and each $k=1,\cdots, 2^{n-1}$, let

$$B_{n,k} = \left \{(x,y)\in \Bbb R^2: \left (x-\frac{2k-1}{2^n}\right )^2 + y^2 =4^{-n} \mbox{and} y\ge 0\right \};$$

for each $n=0,1,\cdots$ and each $k=1,\cdots, 3^{n}$, let

$$C_{n,k} = \left \{(x,y)\in \Bbb R^2: \left (x-\frac{2k-1}{2\... ...^2 + y^2 =\frac {1}{4}\cdot 9^{-n} \mbox{and} y\le 0\right \};$$

let

$$X = A \cup \left [\bigcup_{n=1}^\infty \left ( \bigcup_{k=1}... ...}^\infty \left ( \bigcup_{k=1}^{3^{n}} C_{n,k}\right )\right ]$$

1) X is an example of an hlc continuum which is not regular.

2) X is the union of two regular continua.

See Nadler 10.59, p.193.

Figure ( A ) Semicircle Continuum


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