Spaces That Are Connected but Not Path-connected

A topological space X is called connected if it’s impossible to write X as a union of two nonempty disjoint open subsets: if X = U ∪ V where U and V are open subsets of X and U ∩ V = ∅ then one of U or V is empty. Intuitively, this means X consists of one piece. A subset of a topological space is called connected if it is connected in the subspace topology. The most fundamental example of a connected set is the interval [0, 1], or more generally any closed or open interval in R. Most reasonable-looking spaces that appear to be connected can be proved to be connected using properties of connected sets like the following [2, pp. 149–151]: • if f : X → Y is continuous and X is connected then f(X) is connected, • if C is a connected subset of X then C is connected and every set between C and C is connected, • if Ci are connected subsets of X and ⋂ iCi 6= ∅ then ⋃ iCi is connected, • a direct product of connected sets is connected.