nlab presents a [proof](https://ncatlab.org/nlab/show/locally+compact+topological+space#categorytheoretic_properties) that the category of locally compact Hausdorff spaces does not admit infinite products in general. In particular it shows that there is no infinite product of $\mathbb{R}$, since such a product would be a topological vector space, by the universal property of the projection maps, and it would also be a product in $\mathbb{R}\text{Vect}$. It is known that every locally compact Hausdorff topological vector space is finite dimensional, see [here](https://terrytao.wordpress.com/2011/05/24/locally-compact-topological-vector-spaces/) for example.


However since linear maps in $\mathbb{R}\text{Vect}$ are not in general proper, such a proof doesn't work in the case that you restrict to proper continuous maps between locally compact Hausdorff spaces.


I wanted to know if there is still a known counterexample that means that this category will not have products?


One reason that suggests to me they might is that topological spaces with local homeomorphisms (étale maps) has products, and they are very weird; for example $\mathbb{R}\times\mathbb{R}^2=∅$ in that category. It also massively restricts what maps can be onto compact spaces (they must come from a compact space, which is another 'intuition' for why it might have products.



Does the category of locally compact Hausdorff spaces with proper maps have products?
Suppose $X,Y$ are locally compact Hausdorff spaces admitting a product $X\otimes Y$ in the category $\mathcal{H}$ of all such spaces. If one of $X,Y$ is empty, then $X\otimes Y$ exists, so I'll assume that neither is.


Then the points of $X\otimes Y$ are exactly the maps $\ast\rightarrow X\otimes Y$. Since $X\otimes Y$ is Hausdorff, any such map is proper. Because of this, both of the categorical projections
$$X\xleftarrow{\pi\_X} X\otimes Y\xrightarrow{\pi\_Y}Y$$
are surjective. They are continuous and proper by assumption.


Working now in $Top$, the category of all spaces, we have the Tychonoff product $X\times Y$ and an induced map
$$\theta:X\otimes Y\rightarrow X\times Y$$
factoring the projections. By the above observation, $\theta$ is surjective.



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> **Lemma**: Suppose that $f:A\rightarrow B$ and $g:B\rightarrow C$ are maps of spaces $A,B,C$. If $f$ is surjective and $g\circ f$ is proper, then $g$ is proper. $\quad\blacksquare$
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Apply the lemma to the composite
$$pr\_X\circ \theta=\pi\_X$$
to conclude that $pr\_X$ is proper. This is only possible if $Y$ is compact. Similarly, $X$ must also be compact.


Of course, if $X,Y$ are compact, then the product $X\otimes Y$ exists in $\mathcal{H}$ and is given by the Tychonoff product $X\times Y$.



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> If $X,Y$ are locally compact Hausdorff and nonempty, then the product $X\otimes Y$ exists in $\mathcal{H}$ if and only if both $X,Y$ are compact.
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