Recall that a topological space \(X\) is locally connected if for every point \(x\in X\) and every neighbourhood \(U\) of \(x\) there exists a connected neighbourhood \(V\) of \(x\) with \(V\subseteq U\).
Prove that a topological space \(X\) is locally connected if and only if for every open set \(U\subseteq X\) the components of \(U\) are open. Now let \(p:X\to Y\) be a quotient map.
Prove that if \(C\) is a component of an open subset \(U\subseteq Y\) then \(p^{-1}(C)\) is a union of components of \(p^{-1}(U)\).
Deduce that if \(X\) is locally connected then so is \(Y\).
Solution
1. If \(X\) is locally connected and \(U\subseteq X\) is open, every component \(C\) of \(U\) is open in \(X\).
Let \(x\in C\). There is a connected neighborhood \(V\) of \(x\) with \(V\subseteq U\). \(V\) is connected, lies in \(U\), and meets \(C\) at \(x\), so \(V\subseteq C\) by maximality of \(C\). \(V\) contains an open set \(W\ni x\), so \(x\) is interior to \(C\).
2. If every component of every open subset of \(X\) is open, then \(X\) is locally connected.
Let \(N\) be a neighborhood of \(x\), containing an open set \(U\ni x\). The component \(C\) of \(U\) containing \(x\) is open by hypothesis and connected, and \(C\subseteq U\subseteq N\).
3. (b) If \(C\) is a component of an open set \(U\subseteq Y\), then \(p^{-1}(C)\) is a union of components of \(p^{-1}(U)\).
Let \(z\in p^{-1}(C)\) and let \(D\) be the component of \(p^{-1}(U)\) containing \(z\). \(p(D)\) is a connected subset of \(U\) containing \(p(z)\in C\), so \(p(D)\subseteq C\) by maximality of \(C\), and \(D\subseteq p^{-1}(C)\). Hence \(p^{-1}(C)\) is the union of the components \(D\) of its points.
4. If \(X\) is locally connected, then every component \(C\) of every open set \(U\subseteq Y\) is open in \(Y\).
\(p^{-1}(U)\) is open, so by step 1 its components are open in \(X\). By step 3, \(p^{-1}(C)\) is a union of them, hence open, and \(C\) is open because \(p\) is a quotient map.
5. Q.E.D.
Steps 1 and 2 prove (a), and step 3 proves (b). For (c), step 4 and step 2 applied to \(Y\) show that \(Y\) is locally connected.