In 1967 Steenrod wrote what later became a quite celebrated paper, A convenient category of topological spaces (Michigan Math. J. 14 (1967) 133–152). The paper conveys the work of many (among the most significant, the PhD thesis of Ronnie Brown) and provides a description of a convenient category of topological spaces.

Among the requirements, we see cartesian closedness. Since then, several papers studying categories of spaces like topoi, or locales, have characterised exponentiable objects, having in mind that seminal paper of Steenrod and it is impossible to recall all the papers impacted by that work.

In a sense, even the gros-toposes used in synthetic differential and algebraic geometry have among their main features to be cartesian closed, connecting the call of Steenrod to the general framework of axiomatic cohesion (and its relatives).

I always found this requirement very useful. Of course many constructions come more natural, or just easier, when one can juggle with an internal hom. Recently though I have been wondering what actually motivated Steenrod, whether he has something very concrete in mind, or was guided from an aesthetic need.

Q. In which proofs, theorems, or main constructions of general or algebraic topology (possibly also algebraic geometry) is it important or very useful that the homset actually carry a structure of topological spaces?

List of applications of internal homs I am aware of.

• Exa. 1. Higher homotopy groups can be reduced to $\pi_0$ via loop-spacing.

• Exa. 2. Similarly cohomology, which can be phrased in terms of cohomotopy.