3
$\begingroup$

The following result is in Topological Groups and Related Structures, Arhangel'skii, Tkachenko;

Notation: $F_a(X)$ denotes the free group on $X$ and $F(X)$ the free topological group on $X$.

Claim: Let $f:X\to Y$ be a contiuous mapping of Tychonoff spaces. Then $f$ admits an extension to continuous homomorphism $F(f):F(X)\to F(Y)$. In addition, if $f$ is a quotient, then $F(f)$ is open.

We recall we already proved that $F(X)$ and $F_a(X)$ are the same of algebraic point of view.

Proof: Since $Y$ is identified with the corresponding subspace of $F(Y)$, we can consider $f$ as a continuous mapping of $X$ to the free topological group $F(Y)$. Therefore, by the definition of $F(x)$, $f$ can be extended to a continuous homomorphism $F(f):F(X)\to F(Y)$ wich we shall denote by $\overline{f}$.

Now suposse that $f$ is a quotient. Denote $\tau _q $ the family of all $\overline{f}(U)$ where $U$ is open in $F(X)$. Then $\tau _q $ is a group topology on the abstract group $F_a(Y)$.

Here are my questions:

1.- Why is $\tau _q$ a topology? I don't see why is closed under finite intersections.

2.- Why is $(F_a(Y),\tau _q)$ a topological group? I don't know how to prove that the map $(x,y)\mapsto xy$ is continuous on $F_a(Y)\times F_a(Y)$.

I recall that $f$ is a quotient map if it is surjective, continuous and $f^{-1}(V)$ is open implies $V$ is open.

Thanks.

$\endgroup$
  • $\begingroup$ Are you the same person as math.stackexchange.com/users/73564/user73564? $\endgroup$ – user17762 May 27 '13 at 4:28
  • $\begingroup$ Yes... I really don't know why I have two different accounts. I think it is because I have two PC's... anyway. By the way, I posted about this result in general topological spaces but it was false... It must be the structures of free groups were necessary or something like that. $\endgroup$ – user73564 May 27 '13 at 4:34
2
$\begingroup$

It seems that $(F_a(Y),\tau_q)$ is a quotient group (see section 1.5 of “Topological Groups and Related Structures” about this construction) of the topological group $F(X)$, and $\bar f$ is the quotient map.

$\endgroup$
  • $\begingroup$ What is that quotient group? $\endgroup$ – user73564 Jun 7 '13 at 0:55
  • $\begingroup$ The group $(F_a(Y),\tau_q)$ should be a quotient group of the topological group $F(X)$ by the definition of the topology $\tau _q$ as the family of all $\overline{f}(U)$ where $U$ is open in $F(X)$. $\endgroup$ – Alex Ravsky Jun 7 '13 at 4:01

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.