Continuous Functions are between Topological Spaces not Sets

In the Poincare Project, I've said things of the form "a continuous function from (some subset of the real numbers)".

There's an assumption in that phrase that's worth pointing out.

Whenever someone talks about a continuous function, they are actually talking about a mapping between topological spaces rather than just between two sets. This is because the definition of "continuous" requires a topology.

So, in this context, whenever I say "the reals", I mean "the topological space consisting of the set of real numbers with the standard order topology". Recall that any totally ordered set has a particular topology that can be derived from the ordering relation.

Mathematicians frequently take this kind of shortcut and it should always be clear from the context what is being referred to. But I think it's useful to point out because I think it's something that needs to be understood explicitly.

UPDATE: next post