Topological Group Reflections: turning a group with topology into a topological group, Part II

In Part I, we discussed the operation $G\mapsto \tau(G)$ , which takes in a group equipped with a topology (that does not necessarily interact nicely with the operations of the group) and outputs a topological group. The underlying group of $\tau(G)$ is the same as $G$ but the topology of $\tau(G)$ is coarser than that of $G$ . The construction ensures that $\tau(G)=G$ if and only if $G$ is a topological group.

In this post we’re going to show that $\tau$ is a functor and consider the consequences of this fact. Before that recall the inductive construction of $\tau(G)$ requires a step-construction $G\mapsto c(G)$ where $c(G)$ is $G$ equipped with the quotient topology inherited by the map $\sigma:G\times G\to G$ , $\sigma(a,b)=ab^{-1}$ . Using this, we start with $G$ with topology $\mathcal{T}_0(G)$ and recursively define $c^{\alpha+1}(G)=c(c^{\alpha}(C))$ for each ordinal $\alpha$ and let $\mathcal{T}_{\alpha+1}(G)$ denote the topology of $c^{\alpha+1}(G)$ . If $\alpha$ is a limit ordinal then $c^{\alpha}(G)$ is the group $G$ equipped with the topology $\mathcal{T}_{\alpha}(G)= \bigcap_{\beta<\alpha}\mathcal{T}_{\beta}(G)$ .

The main idea of the construction is that for any given group with topology $G$ , the transfinite sequence $\{c^{\alpha}(G)\}_{\alpha}$ of groups with topology eventually stabilizes at a topological group $\tau(G)$ .

We consider two categories. Let $\mathbf{GrpwTop}$ will denote the category of groups with topology and where the morphisms are continuous group homomorphisms. Let $\mathbf{TopGrp}$ denote the category of topological groups and continuous homomorphisms. Note that $\mathbf{TopGrp}$ is a full subcategory of $\mathbf{GrpwTop}$ and so we have an inclusion functor $i: \mathbf{TopGrp}\to \mathbf{GrpwTop}$ .

Lemma 1: $c:\mathbf{GrpwTop}\to \mathbf{GrpwTop}$ is a functor.

Proof. We have already defined $c$ on objects. Let $f:G\to H$ be a continuous group homomorphism. We define $c(f):c(G)\to c(H)$ to be the same function $f$ . Once we show that $c(f)$ is continuous, the conditions of being a functor follow easily. Recall that $\sigma_G:G\times G\to c(G)$ , $\sigma_G(a,b)=ab^{-1}$ and $\sigma_H:H\times H\to c(H)$ , $\sigma_H(x,y)=xy^{-1}$ are quotient maps by definition. We have

$c(f)\circ \sigma_{G}(a,b)=f(ab^{-1})=f(a)f(b)^{-1}= \sigma_{H}\circ (f\times f)(a,b)$

and thus $c(f)\circ \sigma_{G}= \sigma_{H}\circ (f\times f)$ .

Since this composition is continuous and $\sigma_G$ is quotient, the map $c(f)$ is continuous by the universal property of quotient maps. $\square$

Theorem 2: For every ordinal $\alpha$ , $c^{\alpha}:\mathbf{GrpwTop}\to \mathbf{GrpwTop}$ is a functor when we define $c^{\alpha}(f):c^{\alpha}(G)\to c^{\alpha}(H)$ to be the homomorphism $f$ on underlying groups.

Proof. This proof is by transfinite induction. Lemma 1 shows that $c^1$ is a functor and if $c^{\alpha}$ is a functor, then so is $c^{\alpha+1}=c\circ c^{\alpha}$ . Thus it suffices to focus on the limit ordinal case and show that $c^{\alpha}(f)$ is continuous when $\alpha$ is a limit ordinal. Our induction hypothesis is that $c^{\beta}$ is a functor for all $\beta<\alpha$ . Thus $c^{\beta}(f):c^{\beta}(G)\to c^{\beta}(H)$ is continuous for all $\beta<\alpha$ . To show that $c^{\alpha}(f):c^{\alpha}(G)\to c^{\alpha}(H)$ is continuous, let $U\in \mathcal{T}_{\alpha}(H)=\bigcap_{\beta<\alpha}\mathcal{T}_{\beta}(H)$ . For fixed $\beta<\alpha$ , we have $U\in \mathcal{T}_{\beta}(H)$ and since $c^{\beta}(f)$ is continuous, $f^{-1}(U)=(c^{\beta}(f))^{-1}(U)\in \mathcal{T}_{\beta}(G)$ . Thus $f^{-1}(U)=(c^{\alpha}(f))^{-1}(U)\in \mathcal{T}_{\alpha}(G)$ . We conclude that $c^{\alpha}(f)$ is continuous. $\square$ .

Theorem 3: $\tau:\mathbf{GrpwTop}\to \mathbf{TopGrp}$ is a functor when we define $\tau(f):\tau(G)\to \tau(H)$ to be the homomorphism $f:G\to H$ on underlying groups.

Proof. We have already defined $\tau$ on objects (see Part I for details). Let $f:G\to H$ be a continuous homomorphism of groups with topology. Find ordinals $\alpha$ and $\beta$ with $c^{\alpha}(G)=\tau(G)$ and $c^{\beta}(H)=\tau(H)$ . Set $\gamma=\max\{\alpha,\beta\}$ . Then $c^{\gamma}(G)=\tau(G)$ and $c^{\gamma}(H)=\tau(H)$ . Since $c^{\gamma}$ is a functor by Theorem 2, we conclude that $c^{\gamma(f)}=\tau(f)$ is continuous. $\square$ .

Using functorality, we can easily verify the “universal property” of the construction $\tau(G)$ .

Theorem 4: If $G$ is a group with topology, $H$ is a topological group and $f:G\to H$ is a continuous group homomorphism, then $f:\tau(G)\to H$ is also continuous.

Proof. Recall that the topology of $\tau(G)$ is coarser than that of $G$ so this is not completely obvious. However, we do know that $\tau(H)=H$ is $H$ is assumed to be a topological group. Therefore, applying the functor $\tau$ to $f$ , we have $f=\tau(f):\tau(G)\to \tau(H)=H$ is continuous. $\square$ .

Corollary 5: The topology of $\tau(G)$ is the finest topology on the underlying group of $G$ , which is (1) coarser than that of $G$ and (2) is a group topology (makes the group operations continuous).

Proof. Suppose $H$ denotes the group $G$ with a topology that is (1) coarser than that of $G$ and (2) a group topology. (2) implies that the identity function $id: G\to H$ is continuous. Using (1), we may apply Theorem 4, which implies that $id:\tau(G)\to H$ is continuous. Thus that topology of $\tau(G)$ is finer than that of $H$ . $\square$ .

Conclusion

Theorem 4 implies that $\tau:\mathbf{GrpwTop}\to\mathbf{TopGrp}$ is a reflection functor in the sense that it is left adjoint to the inclusion functor $i:\mathbf{TopGrp}\to\mathbf{GrpwTop}$ . That is, for a topological group latex $H$ and group with topology $G$ , there is a natural bijection $\mathbf{TopGrp}(\tau(G),H)\cong\mathbf{GrpwTop}(G,i(H))$ that is the identity on homomorphisms of the underlying groups.

This provides a sense in which the construction $G\mapsto \tau(G)$ becomes the most efficient way one can turn a group with topology $G$ (including quasitopological groups). If $G$ is not a topological group and $\mu$ is group multiplcation, then there is some open set $U$ where $\mu^{-1}(U)$ is not open in $G\times G$ . This construction tells us to throw that set $U$ away. Keep throwing away problematic open sets like this until one ends up with only open sets $U$ where $\mu^{-1}(U)$ is open. We had to use transfinite induction to show that such a procedure is possible. That it worked out shows that it is indeed possible to “remove” a smallest number of open sets from the topology of $G$ until a topological group is obtained.