Continuity

A continuous function between topological spaces X and Y is a map from X to Y such that the pre-image of an open set in Y is open in X.

As computability
From domain theory, we think that continuity can be associated with computability.

What lies underneath the assumption of computability? Algorithmic consistency? What lise underneath the assumption of continuity? $$\mathcal{T}$$ as what makes "the topological assumption"?

Says and does
Sometimes we make a distinction between between hypotheses and operators, and sometimes we say that they are continuous with each other.