Topology: Example of a compact set but its closure not compact Can anyone gives me an example of a compact subset such that its closure is not compact please? Thank you.

Here is an example (using the formalism of affine schemes) illustrating Elden's excellent idea.  
Let $k$ be a field and $\mathbb A^\infty_k$ be the affine scheme associated to the polynomial ring over $k$ in infinitely many variables: $$\mathbb A^\infty_k=\operatorname {Spec} (k[T_n|n\in \mathbb N])$$  
For our  example we will consider the open subspace  $$X=\mathbb A^\infty_k\setminus \{\mathfrak m\}$$ where $\mathfrak m$ is the maximal ideal $\mathfrak m=(T_0 ,...,T_n,..)\subset k[T_n|n\in \mathbb N]$
That subspace $X$  has the  point $\eta=(0)$ corresponding to the zero ideal as its generic point, meaning that  the singleton set $\{\eta\}$ is  dense in $X$ : $\overline{ \{\eta\}}=X$ .
Now  $\{\eta\}$ is certainly compact but its closure $X$ is not quasi-compact:
Indeed, $X$ is covered by the family of principal open subsets $(D(T_n))_{n\in \mathbb N}$ but a finite union $\bigcup^N_{n=0} D(T_n)$ can never  cover $X$ because for the prime ideal $(T_0,...,T_N)\in X$ we have  $(T_0,...,T_N)\notin \bigcup^N_{n=0} D(T_n)$.
