Let $$\begin{align*}X=&\left\{(x,y,z)\in \mathbb{R}^3 \mid x^2+y^2=1, 0\leq z \leq 1 \right\}\\&\cup \left\{ (x,y,z)\in \mathbb{R}^3 \mid x^2+y^2\leq 1, z\in \{0,1\} \right\}\end{align*}$$ equipped with the induced topology by the usual topology in $\mathbb{R}^3$. Consider the quotient space obtained by identifying all points of the base of the cylinder $B=\left\{ x^2+y^2\leq 1 \mid z=0 \right\}$. Let $p$ stand for the canonical projection from $X$ to $X/B$. Study the compactness of both $X$ and $X/B$.
I think I am missing something here but I don't know what.
First, the space $X$ is compact. It is closed and bounded or it can be written as the product $\mathbb{S}^1 \times [0,1]$ union the closed discs $x^2+y^2\leq 1,\ z=0,1$. But then $p$ is a continuous surjection, so $X/B$ is also compact (in the quotient topology).
I've had always trouble when quotient topology questions appear, so I wouldn't be surprised if I missed something. Thanks in advance!