Criteria for dense open subset of schemes The purpose of this question is to expand on a remark made in Definition 7.1.27 in Qing Liu's book. All of what is discussed below is answered in the posts mentioned at the end of this question.   

Proposition. Let $ X $ be a scheme and $ U $ an open subset of $ X $. Consider the following statements.
  1) $ U $ is dense in $ X $.
  2) $ U $ contains all generic points of $ X $.
  3) $  \text {codim} ( X - U , X )  \geq    1    $.
  Then, $ 3)  \Longleftrightarrow 2)  \implies 1) $. If, moreover, $ X $ is locally Noetherian,   these conditions are equivalent.

Related: Multiplicity of Cartier Divisor, Open dense subsets of schemes, Irreducible closed subsets correspond to points, Codimension 1 points and   Relation between points of codimension 1
 A: We need a Lemma first (which is exercise $ 2.5.2 $ in Qing Liu's book). 

Lemma. Let $ X $  be a scheme and $ x \in X $. Then, $  \text{codim} (  \overline {  \left \{ x \right \} } , X ) = \dim \mathcal{O}_{X,x} $. If $ Z $ is a closed subset of $ X $, then $  \text{ codim} ( Z, X ) = \min _ { z \in Z }  \dim  \mathcal{O}_{X,z}  $.  

Proof.  If   $ \left \{ x \right \}  \subseteq  Z_{0}  \subsetneq Z_{1}  \ldots,  \subsetneq Z_{n}  =  X $ is a sequence of irreducible closed subsets of $ X $, then, for any open $ U \subset X $ containing $ x $, we have $     \left \{ x \right \}   \subseteq U \cap  Z_{0}  \subsetneq U \cap Z_{1}    \subsetneq \ldots  \subsetneq    U \cap  Z_{n} = U $. We may thus assume that $ U $ is the spectra of a ring $ A $, and $ x  =  \mathfrak{p} $ is a prime. Then, $   \overline {  \left \{ x  \right \}   }  = V ( \mathfrak{p} )  $ and $   \text { codim } ( V( \mathfrak{p}  )  , X )  =  \text{ht} (  \mathfrak{p} ) =   \dim  A_{ \mathfrak{p} } =     \dim  \mathcal{O} _ { X , x }  $.   
Let $ Z $ be a closed subset of $ X $. Let $ C_{i} $ be the irreducible closed subsets of $ Z $. Then, by definition, $   \text{codim} ( Z, X )   =   \inf  _{ i  }    \text{  codim }  ( C_{i} , X ) $. It therefore suffices to assume that $ Z $ is irreducible. Moreover, if $ U_{j} $ is an open covering of $ Z $, then $   \text{codim} ( Z, X ) =   \inf  _{  j }   \text{codim } ( Z \cap U _{ j } , U _{ j }  )     $  for similar  reasons as above. We may thus also assume that $ X $ is affine and $ Z $ is an irreducible closed subset i.e. $ X =  $ Spec $ A $ and $ Z = V ( \mathfrak { p } ) $ for some prime $ \mathfrak  { p } $ of $ A $. Then, $ \text {codim} ( Z, X ) =  \text{ht} (  \mathfrak{p} ) =  \dim A_ { \mathfrak { p }  } =   \min _ { \mathfrak{q}  \supset \mathfrak{p} }  \dim  A_{\mathfrak{q} }   =   \min _ { z \in Z }  \dim    \mathcal{ O } _ {  X , z }   $.  $ \quad \quad \quad    \quad   $         $  \square  $
We now turn to the proof of the proposition.
 $   \boxed { 3)  \Longleftrightarrow 2)  }  $.  By the Lemma, we see that $   \text{codim} (  X - U , x ) \geq   1  $ iff $ U $ contains points all the points of codimension $ 0 $, which are the same as generic   points.    
$  \boxed {  2) \implies 1 )  }  $ If $ U $ contains all generic points, then any closed subset of $ X $ that contains $ U $ must contain all the generic points, and thus, must contain all the irreducible components of $ X     $, and  thus must equal $ X $. 
Suppose now that   $ X $ is locally Noetherian. 
$   \boxed {  1)  \implies  2)     }  $ Suppose that $  \eta $ is a generic point of $ X $ outside $ U $. Then,  $  U  \cap   \overline  { \left  \{  \eta   \right \}  }   =  \emptyset $.  Consider the  set $ S  = \left \{ Z | Z \text{ irreducible component of } X \right \}  \setminus     \overline {    \left \{  \eta \right \} }  $. Let  $$  C  =  \bigcup _ { Z \in S } Z   . $$
We claim that   $  C  $ is closed. This will result in a contradiction since $ C \supset U $ and $ C \neq X $.  Let $ V  $ be an open affine subset of $ X $. Then, since $ V $ is Noehterian, the union $   \bigcup _ { Z \in S }  V \cap Z   $   is a finite union  of   closed  subsets of $  V $, and is thus closed in $ V $. Thus, $ C $ is closed in $ X $.   
