I think there is a problem with the following exercise from Zorich, Mathematical Analysis II - Exercise 1 in Section 9.4.1:

Show that in terms of the ambient space the property of connectedness of a set can be expressed as follows: A subset $E$ of a topological space $(X,\tau)$ is connected iff there is no pair of open subsets $A,B$ that are disjoint and such that $E\cap A\neq \emptyset$, $E\cap B\neq \emptyset$, and $E\subseteq A\cup B$.

Thanks in advance!

Edit: counterexample: $X=\{a,b,c\}$, with topology generated by $\{a,b\}$ and $\{b,c\}$, with $E=\{a,c\}$.

  • $\begingroup$ Which problem? Please be specific. $\endgroup$ – Did Dec 14 '18 at 15:07
  • $\begingroup$ @Did I think the conclusion is wrong $\endgroup$ – Jiu Dec 14 '18 at 15:08
  • $\begingroup$ Do you have a counterexample? BTW, when it speaks of open subsets $A, B$, those are meant to be open subsets of $X$. $\endgroup$ – John Hughes Dec 14 '18 at 15:09
  • $\begingroup$ Please say why you think it is wrong $\endgroup$ – MPW Dec 14 '18 at 15:13
  • $\begingroup$ @JohnHughes Yes. $X=\{a,b,c\}$, with topology generated by $\{a,b\}$ and $\{b,c\}$, with $E=\{a,c\}$. $\endgroup$ – Jiu Dec 14 '18 at 15:13

Yes, it's a mistake. It should've been: "$E\cap A$ and $E\cap B$ are disjoint" instead of "$A$ and $B$ are disjoint".


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