Homeomorphism between a cone in a compact space and this compactification Let $X$ be a compact Hausdorff space. Show that the cone in $X$ is homeomorphic to the compactification of $X \times [0,1)$. If $A$ is closed in $X$, show that $X/A$ is homeomorphic to the compactification of $X\setminus A$.
I don't know even how to begin to solve this question, it's seems so hard, anyone could help me, please.

Here’s a start on the second problem; you should be able to use some of the ideas to help you with the first, as well.
Suppose that $X$ is a compact Hausdorff space, and $A\subseteq X$ is closed. Let $Y=X\setminus A$, and let $Y^*=Y\cup\{p\}$ be the one-point compactification of $Y$. Finally, let $Z=X/A$, let $q:X\to Z$ be the quotient map, and let $a\in Z$ be the point corresponding to $A$ in $X$. You want to prove that $Y^*$ is homeomorphic to $Z$.
The first step is figure out what the homeomorphism should be. $Z=q[Y]\cup\{a\}$, where $q[Y]$ is a homeomorphic copy of $Y$, and $Y^*=Y\cup\{p\}$, where $Y$ is a homeomorphic ‘copy’ of $Y$ sitting inside $Y^*$. Thus, each of the spaces $Z$ and $Y^*$ consists of a copy of $Y$ together with one extra point. This suggests that we should try the function
$$h:Y^*\to Z:y\mapsto\begin{cases}
q(y),&\text{if }y\in Y\\
a,&\text{if }y=p\;,
\end{cases}$$
which sends each point of $Y$ to its copy in $Z$ and sends the extra point $p$ in $Y^*$ to the extra point $a$ in $Z$. Now you just have to check that this $h$ really is a homeomorphism.
