On which topological spaces, can we give a group structure to make it a topological group? Let $X$ be a non-empty set. It is known that we can give a group structure on $X$. Now let $X$ be a non-empty topological space. Then can we give a group structure on $X$ so that it becomes a topological group w.r.t. its original topology ?
 A: No, you cannot do that for all spaces $X$.
If $X$ has the structure of a topological group, it implies a lot of extra facts about it, and those give necessary conditions that $X$ should fulfill.
Some examples of such properties: 


*

*If $X$ is $T_0$ it must also be $T_{3\frac{1}{2}}$ (Tychonoff). (it's uniformisable)

*$X$ is homogenous: for every $x, y \in X$ there is a homeomorphism $h:X \to X$ such that $h(x) = y$.

*$X$ does not have the fixed point property (any non-unit multiplication shows this)

*If $X$ is compact it is dyadic and thus ccc. 

*If $X$ is first countable and $T_0$ it is metrisable. (Birkhoff metrisation theorem).


So e.g. $X= [0,1]^n$ cannot be made into a topological group, because of both 2 and 3. The Sorgenfrey line fails 5. The infinite cofinite topology fails 1. 
So many spaces cannot have a structure of a topological group. 
@orangeskid mentioned an algebraic topology reason of possible failure: $\pi_1(X)$ is Abelian when $X$ is a topological group. This makes the wedge sum of circles $S^1 \vee S^1$ another example, I believe.
