What is π_1(BG) for an arbitrary topological group $G$? The classifying space $BG=|Nerve(G)|$ of an arbitrary topological group $G$ does not necessarily have the homotopy type of a CW-complex but the fundamental group should still be accessible. What is $\pi_{1}(BG)$? A reference on this would be great. My initial guess: $\pi_{1}(BG)$ is the quotient group $\pi_{0}(G)$ for arbitrary $G$
Motivation: There is a natural way to make $\pi_1$ a functor to topological groups. I am interested in relating the topologies of $G$ and $\pi_{1}(BG)$ but the topology on $\pi_{1}(X)$ is boring (discrete) when $X$ is a CW-complex.
 A: If $G$ is homeomorphic to a Cantor set (e.g. $G=\mathbb Z_p$), then $BG$ contains a copy of the Hawaiian earrings in it. To see this, take a sequence of points of $G$ that converges to the identity element: you'll get a corresponding sequence of 1-cells in $BG$ that converge to the the degenerate 1-cell. The fundamental group of the Hawaiian earrings is a rather wild object, and looks nothing like the free group that you might naively expect.
If, on the other hand, if you agreed to redefine $BG$ to be the fat geometric realization of the simplicial space $NG$, then you would get $\pi_1(BG)\cong\pi_0(G)$, as desired.
I would even bet that the above isomorphism respects the natural topologies that are present on both sides.
A: Readers might also like to know a result on the first k-invariant of BG contained in
R. Brown and C.B. Spencer,  $\cal G$-groupoids, crossed modules
and the  fundamental groupoid of a topological group'', Proc.
Kon. Ned. Akad. v.  Wet. 7 (1976) 296-302.
This shows that the associated crossed module to fundamental groupoid of $G$ has   $k$-invariant which is exactly the $k$-invariant of $BG$. 
Since this query is about non-connected topological groups, another related ressult is on universal covers of non-connected topological groups 
R. Brown and O. Mucuk, ``Covering groups of non-connected
topological  groups revisited'',  Math. Proc. Camb. Phil.
Soc,  115 (1994) 97-110.
which relates the question of the existence of topological group universal covers of $G$ to the theory of ostructions to extensions of abstract groups. 
A: The first reference in this general area was:

N. E. Steenrod, "Milgram's classifying
  space of a topological group", Topology
  7 (1968) 349–368.

Working in the category of compactly generated Hausdorff spaces, and
combining his Theorems 8.1 and 8.3 we have:

Theorem: Let $G$ be a topological group with
  unit $e$, such that $(G, e)$ is an NDR.
  Then the canonical
  map $EG \to BG$ is a
  quasi-fibration and a principal
  $G$-bundle.

The NDR-condition asks that the inclusion of $e$ in $G$ is
a cofibration.  That is of course weaker than asking for $G$ to be
a CW-space, but not as general as how the question was posed.
Since $EG$ is contractible, the long exact sequence in homotopy
for a quasi-fibration, at the usual base point of $BG$, gives a bijection
$$
\partial : \pi_1(BG) \cong \pi_0(G)
$$
of $\pi_1(BG)$-sets.  This at least gives you an isomorphism
between $\pi_1(BG)$ and some group structure on $\pi_0(G)$.
