Timeline for (A kind of) Irreducibiliy of regular open convex sets in the Cartesian space

8 events

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Jul 23, 2015 at 12:45	comment	added	Eric Wofsey		Given any point $y$ in $V$, if you draw a line segment from $y$ to $x$, and then extend the line segment a little bit past $x$, you will enter the set $W\cdot U$. Thus $x$ lies on the line segment joining $y$ to some point in $W\cdot U$. Since $W\cdot U\leq C$ and $C$ is convex, $y\in C$ would imply $x\in C$.
Jul 23, 2015 at 12:41	comment	added	Rafał Gruszczyński		Eric, I have some second thoughts about the proof. Could you please explain to me the transitions in the following passage: "If $y\in V\cdot C$, then the line from $y$ to $x$ extends into $W\cdot U$, so by convexity $C$ would have to contain $x$". I am not sure if I have understood it properly.
Jul 17, 2015 at 9:10	history	bounty ended	Rafał Gruszczyński
Jul 17, 2015 at 9:10	vote	accept	Rafał Gruszczyński
Jul 17, 2015 at 9:10	comment	added	Rafał Gruszczyński		Ah, OK - I misinterpreted <i>dense in</i> property. Thank you very, very much!
Jul 16, 2015 at 6:58	comment	added	Eric Wofsey		That's just the definition of $D+E$: it is the set of points which have a neighborhood in which $D\cup E$ is dense.
Jul 15, 2015 at 22:13	comment	added	Rafał Gruszczyński		Eric, where does existence of an open ball $U$ in which $D\cup E$ is dense come from?
Jul 15, 2015 at 13:38	history	answered	Eric Wofsey	CC BY-SA 3.0