Consider the lower limit topology on IK and the topology given by the basis  of Exercise o...

Consider the lower limit topology on IK and the topology given by the basis  of Exercise of §13. Determine the closures of the intervals  and  in these two topologies.

(a) Apply Lemma 13.2 to show that the countable collection

is a basis that generates the standard topology on

(b) Show that the collection

is a basis that generates a topology different from the lower limit topology on