I am wondering if there is a topological version of affine spaces as a topological space along with a free transitive (continuous) action of a topological vector space on it?

Here is a notion so-called "Topological affine spaces" which is different from what mentioned above.

  • $\begingroup$ Actually If I'm correct this other notion was invented by M. Fréchet in 1926 as "espace topologiquement affine", which was mistranslated as "topological affine space", while the correct translation would have been "topologically affine space". I've found a few occurrences of "espace afffine topologique" in the meaning you're looking for. $\endgroup$ – YCor May 10 at 6:47

Take any affine space $(A,\vec A)$, with the vector space $\vec A$ endowed by a topology $T_{\vec A}$ making $\vec A$ a topological vector space. Fix any $a_0\in A$ and let $$T_A:=\{a_0+V\colon V\in T_{\vec A}\}, $$ where $a_0+V:=\{a_0+v\colon v\in V\}$. Then $T_A$ is a topology over $A$ (because the map $\vec A\ni v\mapsto a_0+v\in A$ is a bijection). Moreover, the action $$A\times\vec A\ni(a,v)\mapsto a+v\in A$$ of $\vec A$ on $A$ is continuous, as desired.

Details on the latter sentence: Take any $a_0+V\in T_A$, so that $V\in T_{\vec A}$. Take then any $(a,v)\in A\times\vec A$ such that $a+v\in a_0+V$. By the transitivity of the action of $\vec A$ on $A$, there is some vector $v_a\in\vec A$ such that $a=a_0+v_a$. By the associativity of the action, $a_0+(v_a+v)=(a_0+v_a)+v=a+v\in a_0+V$, whence $$v_a+v\in V.$$ So, by the continuity of the addition in the topological vector space $\vec A$, there are members $V_1$ and $V_2$ of the topology $T_{\vec A}$ such that $0\in V_1$, $0\in V_2$, and $$v_a+V_1+v+V_2\subseteq V.$$ Using now the associativity of the action again, we have \begin{equation} (a+V_1)+(v+V_2)=a_0+(v_a+V_1+v+V_2)\subseteq a_0+V. \tag{1} \end{equation} Since $0\in V_2$ and $0\in V_1$, we have $v\in v+V_2\in T_{\vec A}$ and $a\in a+V_1$. Also, $a+V_1=a_0+(v_a+V_1)$ and $v_a+V_1\in T_{\vec A}$, so that $a\in a+V_1\in T_A$. Thus, (1) shows that the action $$A\times\vec A\ni(a,v)\mapsto a+v\in A$$ of $\vec A$ on $A$ is indeed continuous.

  • $\begingroup$ Is there any survey paper or book talking about this. $\endgroup$ – Alireza Ahmadi May 7 at 5:07
  • $\begingroup$ @ARA : I think this is too natural and straightforward to talk about it in a publication, except for giving it maybe a couple of lines. I'd be surprised if this were otherwise. $\endgroup$ – Iosif Pinelis May 7 at 12:09

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