Boxes Continuum

Boxes Continuum is a nice continuum.

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

for each $n=0,1,2,\cdots$ and each $m=0,\cdots, 2^{n+1}$, let

$$B_{n,m} = \left \{(m\cdot 2^{-n-1},y)\in \Bbb R^2: 0\le y \le2^{-n}\right \};$$

finally, let

$$X = \left [\bigcup_{n=1}^\infty A_n\right ] \cup \left [\big... ...infty \left ( \bigcup_{m=0}^{2^{n+1}} B_{n,m}\right )\right ].$$

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

2) X is the union of two regular continua.

See Nadler 10.38, p.186.

Figure ( A ) Union of two dendrites, a non-hlc Boxes Continuum


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