How are these equivalent to the regular epsilon-delta definition of continuity?
(i) Let . The pre-image of each closed set in is closed in .
(ii) The pre-image of each open set in is open in .
In a general space they are not equivalent. But they are in a metric space and in particular .
Note that .
In the basic open set is an open inteveral. is the a basic open inteveral, in fact it is a ball centered at with radius .