# Solution: 2019-10 Is there canonical topology for topological groups?

Let $$G$$ be a group. A topology on $$G$$ is said to be a group topology if the map $$\mu: G \times G \to G$$ defined by $$\mu(g, h) = g^{-1}h$$ is continuous with respect to this topology where $$G \times G$$ is equipped with the product topology. A group equipped with a group topology is called a topological group. When we have two topologies $$T_1, T_2$$ on a set S, we write $$T_1 \leq T_2$$ if $$T_2$$ is finer than $$T_1$$, which gives a partial order on the set of topologies on a given set. Prove or disprove the following statement: for a give group $$G$$, there exists a unique minimal group topology on $$G$$ (minimal with respect to the partial order we described above) so that $$G$$ is a Hausdorff space?

The best solution was submitted by 이정환 (수리과학과 2015학번). Congratulations!

Here is his solution of problem 2019-10.

An incomplete solutions were submitted by 채지석 (수리과학과 2016학번, +2).

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Let $$G$$ be a group acting by isometries on a proper geodesic metric space $$X$$. Here $$X$$ being proper means that every closed bounded subset of $$X$$ is compact. Suppose this action is proper and cocompact,. Here, the action is said to be proper if for all compact subsets $$B \subset X$$, the set $\{g \in G | g(B) \cap B \neq \emptyset \}$ is finite. The quotient space $$X/G$$ is obtained from $$X$$ by identifying any two points $$x, y$$ if and only if there exists $$g \in G$$ such that $$gx = y$$, and equipped with the quotient topology. Then the action of $$G$$ on $$X$$ is said to be cocompact if $$X/G$$ is compact. Under these assumptions, show that $$G$$ is finitely generated.