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We have the following theorem (from Husain's Introduction to Topologcal Groups), slightly rephrased:

The following are true (apologies for the sloppy formatting):

(1) in any topological group $G$, there is a local base $\mathscr U_e$ at $e$ consisting of closed sets such that:

(a) $U=U^{-1}$ for all $U \in \mathscr U_e$

(b) For every $U \in \mathscr U_e$, there is $V \in \mathscr U_e$ such that $VV \subseteq U$.

(c) For every $U \in \mathscr U_e$ and every $a \in G$, there is $V \in \mathscr U_e$ such that $V \subseteq a^{-1} Ua$.

(2) Conversely, for a plain group $G$, let $\mathscr B$ be a filterbase satisfying (a)-(c) above. Then there is a unique topology on $G$ making $G$ into a topological group such that $\mathscr B$ is a local base at $e$.

We can canonically make any topological group $G$ into a uniform space as follows (Willard, Problem 35F): take a local base $\mathscr U$ at $e$ consisting of symmetric nbhds. Then, the left uniformity $\mathscr D_L$ is the uniformity on $G$ generated by the base $\{L_U | U \in \mathscr U\}$, where $L_U=\{(x,y) \in G^2 | y \in xU\}$. One can do the same for right uniformities, but I will restrict to only left uniformities here.

My (perhaps vague) question is as follows: can we rephrase the theorem stated above in terms of the canonical uniform structure on a topological group, or is it merely coincidence that conditions (a) and (b) resemble rather strongly some of the axioms for a uniform space?

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Secondly, I think it is worth mentioning the following theorem from Bourbaki:

The following are true:

(1) Let $E$ be a topological module over a valued-division ring $K$. Then, there is a local base $\mathscr B$ of closed nbhds at $0$ satisfying:

(a) Every $V \in \mathscr B$ is balanced and absorbing

(b) If $V \in \mathscr B$ and $\lambda \in K, \lambda \neq 0$, then $\lambda V \in \mathscr B$

(c) For each $V \in \mathscr B$, there is $W \in \mathscr B$ such that $W + W \subseteq V$.

(2)Conversely, given a module over a valued-division ring $K$: for any filterbase $\mathscr B$ satisfying (a)-(c), there is a unique topology on $E$ which makes $E$ into a topological module with $\mathscr B$ a local base at $0$.

I am also interested if one can rephrase this in terms of the natural uniform structure on a topological module (that is, the uniform structure on $E$ if we regard $E$ as an abelian group with respect to addition). It is not clear how to capture conditions (a) and (b) in the language of uniform spaces, however condition (c) seems analogous to the axiom in a uniform space "for all entourages $V$, there is an entourage $W$ such that $W \circ W \subseteq V$".

Ideally, there is some neat way to rephrase both theorems in terms of uniform spaces, since there seems to be quite some resemblance. Any thoughts are welcome. Understandably, my questions are vague, so I'd be happy to clarify.

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  • $\begingroup$ At least in the TVS world the textbooks do talk about invariant pseudonorms (or "F seminorms") which is another way of talking about the neighborhood filter at 0 via the uniform structure it ought to generate. $\endgroup$ – Dap Feb 10 at 15:41
  • $\begingroup$ Do you perhaps have a reference for what you mentioned above, about F seminorms and the uniform structure they generate? I’ve looked in Jarchow’s Locally Convex Spaces and Bourbaki, though I haven’t really found anything. $\endgroup$ – LinearOperator32 Feb 10 at 20:56
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For topological groups, a reference is Bourbaki, General Topology, IX§3. The setting is a uniform structure on $G$ generated by a family of left-invariant pseudometrics.

For topological vector spaces, a reference is Eric Schechter, Handbook of Analysis and Its Foundations, Theorem 26.29:

26.29. Theorem (Birkhoff, Kakutani, and Minkowski). Let $\mathcal T$ be a topology on an Abelian group $X.$ Then (i) [...] (ii) $\mathcal T$ is a TVS topology if and only if $\mathcal T$ is the gauge topology determined by some gauge consisting of $F$-seminorms, and (iii) [...]

The $VV\subseteq U$ or $W+W\subseteq V$ properties are playing two roles, by ensuring joint continuity of the group operation while also ensuring that the filter actually corresponds to the neighbourhood filter of a topology (as in the "$V\circ V\subseteq U$" divisibility axiom for entourages). I would say it's a bit of a coincidence or trick, but that's a matter of opinion.

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