TITLE: Is the empty set a topological space?
QUESTION [5 upvotes]: If so, is the empty function  from it to any other space  considered a continuous function? I can't really convince myself either way.

REPLY [5 votes]: Yes, and Yes.
In all topological spaces the empty set and the space itself are open, so the topological space of the empty set which is the space itself is open.  No problems there.
The empty function $f: \emptyset \rightarrow A$ maps the open set $\emptyset$ to ... $\emptyset \subset A$  Which is open. And the preimage of any open set of A is the open set $\emptyset$ so it is continuous.