In 1967 *Steenrod* wrote what later became a quite celebrated paper, **[A convenient category of topological spaces][1]** *(Michigan Math. J. 14 (1967) 133–152)*. The paper conveys the work of many (among the most significant, the [PhD thesis][2] of *Ronnie Brown*) and provides a description of a [convenient category of topological spaces][3].

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 the 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][4] used in synthetic [differential][5] and algebraic geometry have among their main features to be cartesian closed, connecting the call of Steenrod to the general framework of [axiomatic cohesion][6] (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?


  [1]: https://projecteuclid.org/journals/michigan-mathematical-journal/volume-14/issue-2/A-convenient-category-of-topological-spaces/10.1307/mmj/1028999711.full
  [2]: https://ora.ox.ac.uk/objects/uuid:3af55800-4be7-462f-b91d-9769a6dac2c4
  [3]: https://ncatlab.org/nlab/show/convenient+category+of+topological+spaces
  [4]: https://ncatlab.org/nlab/show/big+and+little+toposes
  [5]: https://en.wikipedia.org/wiki/Synthetic_differential_geometry
  [6]: https://ncatlab.org/nlab/show/cohesive+topos