I have a few quibbles about the nature of the ordering $(a,b)$ versus $(b,a)$ when it comes to membership of an entourage in a uniform space. The Wikipedia article on uniform spaces nowhere asserts that $(a,b)\in V\iff(b,a)\in V$, but rather states that only some entourages will have this property, and they are called symmetric entourages. We can see that for any $V$ an entourage, two symmetric entourages may be generated; $V\cup V^{-1}$ and $V\cap V^{-1}$ are both entourages and are both symmetric, the issue being that although Wikipedia acknowledges asymmetry, I note that it is not mentioned anywhere else that I could find in a problematic way.
First issue:
A net $x_{\bullet}=(x_\alpha)_{\alpha\in A}$ whose domain is $X$ is termed a Cauchy net if $X$ is a uniform space, with uniformity $\Phi$, and the net has the property that $\forall V\in\Phi,\,\exists\gamma\in A:\forall\alpha,\beta\ge\gamma,\,(x_{\alpha},x_{\beta})\in V$. This definition is natural, and innocuous seeming, but there is a fundamental issue (as far as I can tell) in the fact that $(x_\alpha,x_\beta)\in V$ does not imply $(x_\beta,x_\alpha)\in V$. This means that Cauchy nets will have to have the unspoken restriction that their image always lies in a symmetric subset of any entourage. I will comment more about resolutions to this after I raise the next issue.
Second issue:
Judging by the ordering of pairs in this article relating the uniform convergence topologies and the compact-open topology, Wikipedia believes $x$ to be $V$-close to $y$ if $(x,y)\in V$, where $V$ is an entourage on a uniform space which contains $x,y$. The composite $W\circ W$ is defined to be $\{(x,z):\exists y\in X,\,(x,y)\in W\wedge(y,z)\in W\}$. Notice the structure $(x,y),(y,z)$. The article at first adheres to this structure by saying that $(f(x_j),\color{red}{f(y)})\in W\wedge(\color{red}{f(y)},g(y))\in W\implies(f(x_j),g(y))\in W\circ W$, in their first proof.
However, in the second proof they break this structure - they say that $(f(x_j),f(x))\in W\wedge(f(x_j),g(x))\in W\implies(f(x),g(x))\in W\circ W$ despite the fact that we need an alternating placement of the shared term - it surely should have been $(f(x),f(x_j))$. This is a technicality which really cannot be avoided in their proof, as they implicitly define $b$ is $V$-close to $a$, in the notation $b\in V[a]$, when $(a,b)\in V$. It is not the case that $a$ is $V$-close to $b$, so this damages their proof. We actually only have that $(f(x),g(x))\in W\circ W^{-1}$.
I have come up with potential resolutions to these problems:
- When Wikipedia writes $(x,y)\in V$, they really mean to say that either $(x,y)\in V$ or $(x,y)\in V^{-1}$, so that $(x,y)\in V$ becomes an abbreviated notation for "$x$ and $y$ are $V$-close to each other".
- An entourage $V$ is really taken to be either $V\cup V^{-1}$ or $V\cap V^{-1}$ (which?) for these purposes
- A Cauchy net is really defined by $\forall\alpha\ge\beta\ge\gamma,\,(x_\alpha,x_\beta)\in V$, where I have ordered $\alpha\ge\beta$ to avoid the issue of $(x_\alpha,x_\beta)\in V$ not implying $(x_\beta,x_\alpha)\in V$
- The restriction that Cauchy nets must actually converge in only symmetric subsets of an entourage is OK, since $V\cap V^{-1}\subseteq V$ is always a symmetric sub-entourage that the Cauchy net would be required to converge in anyway, so the issue I raise is perhaps not a problem
Side-note: in the uniform space article, they put: $V\circ U:=\{(x,y):\exists y\in X,\,(x,y)\in U\wedge(y,z)\in V\}$. Is this the conventional order as accepted by most authors? I ask this because I note Wikipedia (potentially) was sloppy with ordering as shown above, and the order does seem to matter; $V\circ U\neq U\circ V$ in general.
Question:
Are the issues I'm raising actually issues at all? If so, are the potential resolutions right? How am I to interpret the Wikipedia definition of a Cauchy net (and the ordering issue in their second proof!) in light of these issues?
As is completely standard in such proofs (but not always explicitly written) we take $W$ to be a symmetric entourage such that $W\circ W \subseteq V$. The entourage axioms ensure that this can be done for any entourage $V$.
It’s the analog of taking $\frac{\varepsilon}{2}$ when we’re given $\varepsilon >0$ in a metric proof.
If we do that everything in the linked proof works as advertised.