# Fiber bundle and fibration of classifying space

Let $BG$ is classifying space of $G$ topological group.

If $G$ is any compact group and $H$ is a closed subgroup of $G$, then the inclusion map $i:H\rightarrow G$ induces \begin{equation*} G/H\rightarrow BH\rightarrow BG \end{equation*} a fiber bundle?

If $G$ is any compact group and $H$ is a closed subgroup of $G$, then the inclusion map $i:H\rightarrow G$ induces
\begin{equation*} G/H\rightarrow BH\rightarrow BG \end{equation*} a fibration?

If $G$ is any compact group and $N$ is a closed normal subgroup of $G$, then the quotient map $\pi :G\rightarrow G/N$ induces
\begin{equation*} BN\rightarrow BG\rightarrow B\left( G/N\right) \end{equation*} a fiber bundle?

If $G$ is any compact group and $H$ is a closed normal subgroup of $G$, then the quotient map $\pi :G\rightarrow G/N$ induces \begin{equation*} BN\rightarrow BG\rightarrow B\left( G/N\right) \end{equation*} a fibration?

Hint: I assume $H$ is closed here. if $G$ is a Lie group and $H$ a CLOSED subgroup of $G$, the classifying space of $G$ is base space of the universal bundle $p_G:EG\rightarrow BG$. Remark that $EG$ is characterized by the fact that it is contractible. Thus the quotient of $EH$ by $H$ is the classifying space of $H$. This gives you a fibration $G/H\rightarrow EG/H=BH\rightarrow EG/G= (EG/H)/G= BG$.