Question

Let $\mathbb{R}$ be the real line with Euclidean topology. Prove that every connected subset of $\mathbb{R}$ is an interval.

Answer 1

ID is not an Interval VERA an Let R be the Real line with Euclidean topology. Prove that every connected subset of R is an Interval. sonh Let S of R Suppose s is not an Interval of R. then by interval defined by Between Ixues ZERTS such mat acacy, consider the sets A = 5nl-od, z) and Az = sn (za) ALIA2 are open by definition of the subspace topology ons Niether is empty because they contain x and y respectively. they are disjoint, and their umon is so Since 2¢S. there fore, A1, A2 is a separation of s. it follows by the definition that s is disconnected. then

so contrapositively, every connected subset of R is an intervala