Answer to Question #217091 in Differential Geometry | Topology for Prathibha Rose

Question

Given an example of a connected space which is not pathwise connected .substantiate. Your claim

Answer 1

$\text{We consider the set RK. That is the all open intervals together with intervals of the form} \\\text{(a,b)∖K where K={$\frac{1}{n}:n∈ \mathbb{Z}^+$} This space is connected as there are no open sets B} \\\text{ and C such that $RK\subset B\cup C, RK \cap B \neq \empty,RK \cap C\neq, RK \cap B \cap C = \empty$ but not}\\\text{path-connected as there }\\\text{is no continous path lying in the set joining 2 arbitrary points in the set}$