# Set boundary preserved by an infinite union

Suppose I have a subset $U\subset\mathbb R^2$ and a real number $r>1$ with the following properties:

1. $U$ is compact;
2. $U\subset rU$ (self-similarity);
3. $0\in U$;
4. there exists an open set $H\subset \mathbb R^2$ such that $0\in\partial H$ (boundary) and $H\cap U=\emptyset$.

Let $V:=\bigcup_{n\geq0} r^n U$.

Question: Whether exists $H_1$ open such that $0\in\partial H_1$ and $H_1\cap V=\emptyset$.

Of course, if the answer is "yes", I would like to see a way how to prove it (it needn't be a complete proof, just some crucial hint).

-

Of course, $H$ (if it exists) can always be taken to be $\mathbb R^2\setminus U$. – Hagen von Eitzen Nov 26 '12 at 18:39
Of course, and the question is: is still $0\in\partial H\setminus V$ and $H\setminus V$ open? – tohecz Nov 26 '12 at 18:40

## 1 Answer

No.

Consider $U=\{(x,y)\mid x^2+(y-1)^2\le 1\lor x^2+(y+1)^2\le 1\}$ (union of two touching closed disks) and $r=2$. Then $V$ is dense in $\mathbb R^2$.

-

damn, a good one! Now I have to find what other good properties does my set $U$ have. – tohecz Nov 26 '12 at 18:39
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 21, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9255321621894836, "perplexity_flag": "head"}