I tried to have an example of a set $S\subset\mathbb{R}^2$ such that $S$ is connected but int$(S)$ is not. Can anyone give me an example and prove it? Thanks
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How about ${(x,y)\in\mathbb R^2\mid xy\ge 0}$? Or two disjoint discs joined by a line? – hmakholm left over Monica Oct 21 '15 at 06:33
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HINT : consider $S$ to be the union of two closed discs that touch at a point on their respective boundaries. This is connected but its interior is not.
R_D
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In this example, $S$ is path connected and hence connected. Can you say why there is always a path between any two points of $S$? – R_D Oct 21 '15 at 07:04
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I can imagine that there is always a path between any two points of $S$. However, this question is designed for those people who don't know what path connected is. That's why I failed to prove $S$ is connected. – Kelan Oct 21 '15 at 07:11