Short version of the question:
Is $\pi_0(O(3)/C2) = Z_2$ as $\pi_0(O(3)) = Z_2$? Here $C_2$ is a cyclic group of order 2.
Long version or the question:
The zeroth homotopy group describes the connectness of a space. Take the space $O(3)/H$ as an example, where $H$ is a subgroup of the Lie group $O(3)$. If $H = \{1\}$, then $O(3)/H = O(3)$ containing two disjoint copies of $SO(3)$. Therefore, $\pi_0(O(3)/H) = Z_2$ in this case.
Then a question occurred to me: does $\pi_O$ in general given by the number of disjoint copies of a space. If I consider a situation $H=C_2$ where $C_2$ is the order-2 cyclic group $\{1,e^{i\pi}\}$. $C_2$ will define an equivalence in each copy of $SO(3)$ by a $\pi$-rotation of some axis, thus the space $O(3)/C_2$ contains two disjoint copies as $O(3)$, is $\pi_0(O(3)/C_2) = \pi_0(O(3))=Z_2$?
Furthermore, if $H = C_i = \{1,-1\}$, $C_i$ defines a map between the two disjoint copies of $SO(3)$, if so does the space $O(3)/C_i$ contain only one connected copy, and therefore $\pi_0(O(3)/C_i) $ is trivial?
These may be quite naive questions, but I only have a background of physics and the references I read do not explain the $\pi_0$ much. Can any one give me some hits? Any references would be much appreciated as well?
$\pi_0$ is not in general a group. It is the set of homotopy classes of maps from a point into the space. Since a homotopy between two constant maps is a path, this is essentially the set of path components of the space (maximal path connected subsets). If the space is a topological group, then the set of path components has a group structure, but this is a special case.