Consider the “infinite broom” \(X\) pictured in Figure 25.1 of the text. Show that \(X\) is not locally connected at \(p\), but is weakly locally connected at \(p\). [Hint: Any connected neighborhood of \(p\) must contain all the points \(a_i\).]
Solution
1. Definition of the space \(X\) (the infinite broom):
1.1. In \(\mathbb{R}^2\), let \(a_n = (1, 1/n)\) for \(n \in \mathbb{Z}_+\), let \(L_n\) be the closed line segment from the origin \((0, 0)\) to \(a_n\), and let \(p = (1, 0)\).
1.2. The space \(X = \left(\bigcup_{n=1}^\infty L_n\right) \cup \{p\}\) is equipped with the subspace topology from \(\mathbb{R}^2\).
definition of \(X\).
1.3. \(X\) is a connected topological space: The union \(\bigcup_{n=1}^\infty L_n\) is path-connected (every segment contains \((0, 0)\)), and \(a_n \to p\) as \(n \to \infty\), so \(p \in \overline{\bigcup_{n=1}^\infty L_n}\).The closure of a connected space is connected, so \(X\) is connected.
standard property of closures of connected sets.
2. Show that \(X\) is not locally connected at \(p\):
2.1. A space \(X\) is locally connected at \(p\) if every open neighborhood \(U\) of \(p\) contains a connected open neighborhood \(V\) of \(p\).
definition of local connectedness at a point.
2.2. Consider the open neighborhood \(U = B(p, 1/2) \cap X\).
\(B(p, 1/2)\) is an open ball in \(\mathbb{R}^2\) disjoint from the origin \((0, 0)\).
2.3. Let \(V \subseteq U\) be any open neighborhood of \(p\) in \(X\).
setup.
2.4. Since \(V\) is open in \(X\), \(V = W \cap X\) for some open set \(W \subseteq \mathbb{R}^2\) containing \(p = (1, 0)\).
subspace topology.
2.5. Since \(a_n = (1, 1/n) \to p\), \(W\) contains \(a_n\) for all sufficiently large \(n \ge N\).
definition of limit in \(\mathbb{R}^2\).
2.6. For each \(n \ge N\), \(V\) contains a non-empty open subsegment of \(L_n\) near \(a_n\).
\(W\) is open and contains \(a_n \in L_n\).
2.7. Since \(V \subseteq U \subseteq B(p, 1/2)\), the origin \((0, 0) \notin V\).
\(\operatorname{dist}(p, (0, 0)) = 1 > 1/2\).
2.8. The subsegments \(V \cap L_n\) for distinct \(n\) do not meet each other or \(p\) in \(V\), so they are clopen in \(V \setminus \{p\}\).Thus \(V\) is disconnected.
the segments \(L_n\) are pairwise disjoint away from \((0, 0)\).
2.9. Hence no connected open neighborhood \(V \subseteq U\) of \(p\) exists, so \(X\) is not locally connected at \(p\).
3 and 8.
3. Show that \(X\) is weakly locally connected at \(p\):
3.1. A space \(X\) is weakly locally connected at \(p\) if every neighborhood \(U\) of \(p\) contains a connected subspace \(C\) of \(X\) such that \(p \in \operatorname{int}(C)\) (or \(C\) is a connected neighborhood of \(p\)).
Munkres §25 definition of weak local connectedness.
3.2. The space \(C = X\) is connected by
1.
1.
1.3. \(X\) is an open neighborhood of \(p\) in \(X\), so \(p \in \operatorname{int}(X) = X\).
the whole space is open in itself.
1.4. Thus \(C = X\) is a connected neighborhood of \(p\), so \(X\) is weakly locally connected at \(p\).
2 and 3.
4. Conclusion: \(X\) is weakly locally connected at \(p\), but not locally connected at \(p\).
2 and 3.
Q.E.D.