It is often said that the category $\sf Top$ of topological spaces and continuous mappings is not cartesian closed. E.g., in the Wikipedia article on compactly generated spaces and in an answer on this site. Can anyone point me at a proof that $\sf Top$ cannot be made into a cartesian closed category? I.e., that there is no way of putting a topology on the space $X\rightarrow Y$ of continuous functions between topological spaces $X$ and $Y$ that makes the natural "Currying" operation from $X \times Y \rightarrow Z$ to $X \rightarrow Y \rightarrow Z$ into a homeomorphism.
Is Top provably not cartesian closed?
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0let us [continue this discussion in chat](http://chat.stackexchange.com/rooms/2894/discussion-between-rob-arthan-and-zhen-lin) – 2012-03-25
4 Answers
Another proof is to show that for the usual function space there is a monoidal closed structure on Top, see for the Hausdorff case
[1] R. Brown, "Ten topologies for $X \times Y$. Quart. J. Math. (2) 14 (1963), 303-319.
[2] R. Brown, ``Function spaces and product topologies'', Quart. J. Math. (2) 15 (1964), 238-250.
See also, and dealing with the non Hausdorff case,
[3] Booth, P.I. and Tillotson, J., "Monoidal closed categories and convenient categories of topological spaces". Pacific J. Math. 88 (1980) 33--53.
These papers give an exponential law $$X^{Z \times_S Y}\cong (X^Y)^Z $$ where the topology on $Z \times _S Y$ is defined by the property that a function $f: Z \times Y \to W$ is continuous if and only if
1 $f|\{z\}\times Y$ is continuous for all $z \in Z$, and
2 $f \circ (1 \times g)$ is continuous for all maps $g: A \to Y$ of compact Hausdorf spaces $A \to Y$,
This product is associative but not commutative, as shown in [1].
The idea of the category of Hausdorff $k$-spaces being "adequate and convenient for all purposes of topology" is mentioned in the Introduction of [1]. For more information see the n-cat-lab.
Read chapter 7 in the second volume of Borceux's Handbook; Proposition 7.1.1 shows some contraints on a monoidal closed structure on $\bf Top$:
- If $U\colon \bf Top\to Set$ is the forgetful functor, then $U\circ(-\otimes-)=U\circ(-\times-)$.
- $U(Y^X)=$the set of continuous maps $X\to Y$.
- The unit for the monoidal structure must be the singleton set.
Proposition 7.1.2 explicitly proves that the category of all top spaces is not cartesian closed.
There is a sketch of a proof at the ncatlab: they define exponentiable spaces in the "examples" section, ie. the spaces for which there is an exponential. Then in the "Counterexamples" section, there is a counterexample in the category of exponentiable spaces. Basically they use local compactness and Hausdorffness to show that the exponential of two topological spaces is not necessarily exponentiable (but read the proof).
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0Thanks. Working back through the references leads to what looks like the original reference, [a paper by Fox](http://www.ams.org/journals/bull/1945-51-06/), and a nice more recent result by [Escardo and Heckman](http://www.cs.bham.ac.uk/~mhe/papers/newyork.pdf) – 2012-03-24