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Definition. Let $(G,\ast)$ be any group. Then $G$ will be said to be a topological group if there exists a topology on $G$ such that the map $f:G\times G\to G$ defined by $f(x,y)=xy^{-1}$ for all $(x,y)\in G\times G$ is continuous.

Now observe that in this definition it is important to note that we first have a group and then the topology which makes $f$ cotinuous. In other words, we impose topological stricture on a group.

My question

  1. Is it possible to impose group structure on a topological space? I don't think that it is possible always. So my question is, if $(X,\tau)$ be a topological space then under which condition(s) does there exists a binary operation $\ast:X\times X\to X$ such that $(X,\ast)$ is a group?

  2. If there is any related research in mathematics literature then can you please let me know about some of those?

It has been pointed out (see below) that this question is a duplicate of this question. However, I don't think that they are for the reason as explained in this comment. It is also not clear to me that imposing a group structure on a topological space (if it's possible of course) is expected to result in a topological group.

  • @LeeMosher: Not sure that this question is an exact duplicate of the linked question. In the question it is explicitly asked that "can we give a group structure on $X$ so that it becomes a topological group", whereas my question simply is "can we give a group structure on $X$". Can you explain to me how they are duplicate? –  Aug 01 '18 at 17:13
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    Are you sure you really mean that you want to completely ignore the given topological structure on the set $X$? Because every nonempty set has a group structure (assuming the axiom of choice), which totally alters the perception of what your question is about... – Lee Mosher Aug 01 '18 at 18:07
  • @LeeMosher: I think I have been a bit imprecise. Suppose that $(X,\text{Cl})$ be a closure space and consider the topology induced from it. What I would like to see is a group structure on $X$ using this $\text{Cl}$ operator, for example? –  Aug 01 '18 at 18:13
  • For example in case of topological group we have a group $G$, then the method of imposing a topological structure on $G$ is to bring into picture the continuity of $f$ as mentioned above. For it we only used the binary operation on $G$. I was expecting something similar to that. Some function from $X\times X$ using only $\text{Cl}$ for example @LeeMosher. –  Aug 01 '18 at 18:18
  • Can anyone please explain why in spite of my pointing out the the linked question is not a duplicate of the current question, it has been closed as a duplicate of that question? –  Aug 02 '18 at 14:14
  • @JohnMa: Would you mind to explain? Not necessarily here but here will also be fine. –  Aug 02 '18 at 14:15
  • To be clear, your question as it is written it is a duplicate. In your comment you attempted to reformulate what you were asking in the language of closure operators, but if you want to do that then it is best to do it in a new question. If you want to try to write a new question along those lines, please feel free to do so. – Lee Mosher Aug 02 '18 at 18:24
  • @LeeMosher: Strangely enough, in my addition to the question, I explained why precisely the question as it is written is not a duplicate (at least not of the linked question). Could you address why that explanation is faulty? (It must be, because otherwise, I see very little point in stating that the question is a duplicate of the linked one.) –  Aug 03 '18 at 04:20
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    I shall give it a try, but I think this will be my last word in this thread. In mathematics, given two categories (in this case, the topological category and the group category) there is a natural problem which arises, namely to form a kind of mix of the two categories. This is what your words mean mathematically, when you write impose topological structure on a group and when you write impose group structure on a topological space. – Lee Mosher Aug 03 '18 at 14:35
  • Although there is no completely natural way to solve this problem when given an arbitrary pair of categories to merge, nonetheless there is an important constraint on the solution: the structures imposed by the two categories should be compatible with each other. For merging the "topological" category and the "group" category into the "topological group" category, the solution which satisfies this constraint turns out to be very natural, namely: the group operation $G \times G \to G$ and the inversion operation $G \mapsto G$ should be continuous. – Lee Mosher Aug 03 '18 at 14:39
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    So, your question is the same as the duplicate question, both of which ask how to merge the topology category and the group category, and it has the same answer, which is the category of topological groups. As I said, if you wish to ask a different question about closure operators rather than topologies, please feel free to do so. – Lee Mosher Aug 03 '18 at 14:40
  • Very interesting comments. Thanks @LeeMosher. –  Aug 03 '18 at 16:01

1 Answers1

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There is a gigantic literature on topological groups, with specialized literature on many special kinds of topological groups. You might start with Lie groups (topological groups which are manifolds). If you like number theory, you will discover that the $p$-adic numbers $\mathbb Q_p$ are a topological group which is locally homeomorphic to the Cantor set.

No, it is not always possible to impose a group structure on a topological space.

For example, a topological group $G$ must be homogeneous which means that for any two points $x,y \in G$ there exists a homeomorphism $L : G \to G$ such that $L(x)=y$: simply use the map $L(g)=f(g,y^{-1}x)$. But there are many nonhomogeneous topological spaces. The first one that comes to mind is $[0,1]$, which has no homeomorphism taking $0$ to $1/2$.

There are even homogeneous topological spaces on which one cannot impose a group structure, in particular many manifolds. You learn in the theory of Lie groups that if a topological group $G$ is a compact manifold then the Euler characteristic of $G$ equals zero. However, there are plenty of compact manifolds with nonzero Euler characterestic, starting with the 2-sphere $S^2$ which has Euler characteristic equal to $2$.

Lee Mosher
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