Is Top provably not cartesian closed?
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).
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.