# Is the following exercise on connected subset wrong?

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 $$E\subseteq A\cup B$$.

Edit: counterexample: $$X=\{a,b,c\}$$, with topology generated by $$\{a,b\}$$ and $$\{b,c\}$$, with $$E=\{a,c\}$$.
• Do you have a counterexample? BTW, when it speaks of open subsets $A, B$, those are meant to be open subsets of $X$. – John Hughes Dec 14 '18 at 15:09
• @JohnHughes Yes. $X=\{a,b,c\}$, with topology generated by $\{a,b\}$ and $\{b,c\}$, with $E=\{a,c\}$. – 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".