Define vector space on sphere! Is it possible to define a vector space on sphere? If i have a space like $X$ homeomorphic to some topological space which is a vector space can i pull back the operation and make $X$ a vector space?
 A: If you're talking about a vector space $V$ over $\mathbb{R}$, with a topology on $V$, such that the operations of addition and scalar multiplication are continuous with respect to this topology, then I don't think there are any nice topologies which will do what you want.  
As a set, any vector space over $\mathbb{R}$ looks like a subset of a product $$\prod\limits_i \mathbb{R}$$ of copies of the reals, consisting of entries which are zero at all but finitely many places.  The notation for such a subset is $\bigoplus\limits_i \mathbb{R}$.  I assume you want all the projection maps $\bigoplus\limits_i \mathbb{R} \rightarrow \mathbb{R}$ to be continuous functions.  In other words, if $\{v_i\}$ is a basis for $V$, and $v_0$ is one of the basis elements, then the function from $V$ to $\mathbb{R}$ given by $\sum\limits_i c_iv_i \mapsto c_i$ is continuous.  
In this case, $V$ must contain the (subspace topology of) the product topology.  It follows that $V$ is not compact.  But the sphere is compact.  So putting a nice vector space structure on the sphere isn't possible.
