I am reading H. Croom's Principles of Topology and in page 139, he gave an example 5.2.5 to show that two points in the same member of any separation of the topological space X, might not belong to the same component of X. In this example, the space X is a subspace of 2 dimensional Euclidean space with the usual topology. X consists of a sequence of line segments converging to a line segment whose midpoint c has been removed. Then [a,c) is one component of X, but why? You see, since the sequence of the segments converge to [a,b], any open set containing [a,c) must intersect with infinitely many segments in the sequence. Therefore, [a,c) must belong to some "bigger" subset of X, which is a component. But why is my reasoning wrong? Thank you!