Question

What are the connected components and the path components of the product space R x R_l where R has the standard topology and R_l has the lower limit topology?

Answer 1

Observe that connected components of topological spaces XxY will be of the form AxB, where A is a connected component of X and Bis a connected component of Y.

is connected with the usual topology. The only connected components of Rl are singletons. So connected components of xRl are exactly of the form x{a} where belongs to .

Now we left with proving The only connected components of Rl are singletons.

To see that the connected components are the singletons, it is enough to show the following:

Lemma: If A⊆Rl is not empty, then A is connected if and only if A is a singleton.

Proof:  Sufficiency is trivial. As for necessity, suppose that A≠∅ and take an arbitrary a∈A. Then, A is the disjoint union of the two (relatively) open sets (−, ) A and [a,)∩A. Since AA is connected, at most one of these open sets can be non-empty, and since a∈[a,∞)∩A ≠ ∅, it follows that (−,a)∩A=∅. Hence, if b∈A, then b∉(−, a), so that b≥a. Since a and b have been arbitrarily chosen, their roles can be interchanged, implying that a≥b. Conclusion: if a,b∈A, then a=b.