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Did I get the definition of connectedness wrong?

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commit 9079bc59db75ea5a52f0891f6b5cbf6cb2a967df 1 parent d472a0e
Mark Jason Dominus (陶敏修) authored
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  1. BIN  topology.pdf
  2. +5 −0 topology.tex
BIN  topology.pdf
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5 topology.tex
@@ -181,6 +181,11 @@ \section*{Connected sets}
that it is connected.\footnote{If you know the definition of $\R$ in
terms of Dedekind cuts, you should be able to see this.}
+% Don't we require a stronger property than I said here, namely that
+% each of the two open sets excludes the closure of the other? In
+% normal spaces my formulation is equivalent. Should I mention this?
+% Maybe in a footnote?
So let's define a connected set: A {\em separation} of a set $S$ is a
partition of $S$ into two nonempty disjoint parts, $X$ and $Y$, such that there
are disjoint open sets $X'$ and $Y'$ with $X\subset X'$ and $Y \subset
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