= Noncompact counterexample to the Šura-Bura lemma
Set $p=(0,0)$, $q=(1,0)$, $\varepsilon_m=1/(m+2)$, and let $A_m$ be the polygonal arc joining $(\varepsilon_m,0)$, $(\varepsilon_m,m)$, $(1-\varepsilon_m,m)$, and $(1-\varepsilon_m,0)$. In the planar subspace $Z=\{p,q\}\cup\bigcup_m A_m$, each arc is <clopen>. The component of $p$ is therefore just $\{p\}$. Any <clopen> neighborhood of $p$ contains the left endpoints of all sufficiently high arcs, hence those entire <connected> arcs, and by closedness also the limit $q$ of their right endpoints. Thus its <quasi-component> is $\{p,q\}$. Higher arcs escape to infinity while their endpoints approach the two distinguished points.
Back to article page
