Let \((X,\mathcal T)\) be a path-connected space and \(\mathcal C=\{U_\alpha\}_{\alpha\in I}\) an open covering of \(X\). Show that for every \(x,y\in X\) there is an \(n\in\mathbb N\) and a sequence of sets \(U_{\alpha_1},\ldots,U_{\alpha_n}\) in \(\mathcal C\) with \(x\in U_{\alpha_1}\), \(y\in U_{\alpha_n}\) and \(U_{\alpha_{i-1}}\cap U_{\alpha_i}\ne\varnothing\) for all \(i=2,\ldots,n\).
Solution
Since \(X\) is path-connected, choose a path \(\gamma\colon[0,1]\to X\) from \(x\) to \(y\).
1. There is a partition \(0=t_0<t_1<\cdots<t_n=1\) and sets \(U_{\alpha_1},\ldots,U_{\alpha_n}\in\mathcal C\) with \(\gamma([t_{i-1},t_i])\subseteq U_{\alpha_i}\) for each \(i\).
\(\{\gamma^{-1}(U_\alpha)\}_{\alpha\in I}\) is an open cover of the compact metric space \([0,1]\), so it has a Lebesgue number \(\delta>0\). Take an equal subdivision with \(t_i-t_{i-1}<\delta\); each \([t_{i-1},t_i]\) then lies in some \(\gamma^{-1}(U_{\alpha_i})\).
2. Q.E.D.
With the sets of step 1, \(x=\gamma(t_0)\in U_{\alpha_1}\) and \(y=\gamma(t_n)\in U_{\alpha_n}\), and for \(2\le i\le n\) the point \(\gamma(t_{i-1})\) lies in \(U_{\alpha_{i-1}}\cap U_{\alpha_i}\).