Answer to Question #217037 in Differential Geometry | Topology for Prathibha Rose

Question

Give an example of a locally connected space which is not connected substantiate your claim.

Answer 1

$\text{The subspace A = [0,$\frac{1}{2})\cup(\frac{1}{2},1]$ of I is locally connected as its components are also }\\\text{connected but it is not connected as we can choose open sets B = (-1,$\frac{1}{2})$ and }\\\text{C =($ \frac{1}{2}$,2) such that A$\subset B \cup C$ and A $\cap B \neq \emptyset$ and A $\cap C \neq \emptyset$ and A $\cap B \cap C = \emptyset$ }$