It is said that the empty set is the initial object of the category of sets, and the empty space is the initial object in the category of topological spaces. So there exists a unique arrow from the initial object to any object in those categories. But I wonder what would be that map?
It seems that at $t = 0$ the map would give an empty set or space, while at $t = 1$ the map would give a non-empty set or space. How is that possible with a continuous map?