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

Question #217037
Give an example of a locally connected space which is not connected substantiate your claim.
1
Expert's answer
2021-07-22T17:56:45-0400

The subspace A = [0,12)(12,1] of I is locally connected as its components are also connected but it is not connected as we can choose open sets B = (-1,12) and C =(12,2) such that ABC and A B and A C and A BC= \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 $\cap B \cap C = \emptyset$ }


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