next up previous contents index
Next: Boxes Continuum Up: New examples Previous: Pseudo-circle

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 \};


X = A \cup \left [\bigcup_{n=1}^\infty \left (
...}^\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

Source files: a.eps . a.gif . a.mws . latex.tex . title.txt .

Here you can read Notes or write to Notes.

Janusz J. Charatonik, Pawel Krupski and Pavel Pyrih