# Definition of dense set

Is this definition is correct?:

Let $\preceq$ an order in $A$, $B \subset A$, with $B \neq \emptyset$. Then $B$ is a dense set in $A$ if $$\forall x,y \in A ( x \prec y \to \exists b \in B( x \prec b \prec y))$$

P.S. $x \prec y$ means $x \preceq y \,\land\, x \neq y$ and $B \subset A$ means $B \subseteq A \,\land\, A \neq B$.

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Are you asking if this definition is correct? –  Avi Steiner Aug 19 '13 at 17:44
@AviSteiner, yes! –  mle Aug 19 '13 at 17:44
@AviSteiner... thansk for corrections! –  mle Aug 19 '13 at 18:32

If $\preceq$ is a linear order, your definition is correct. If $\preceq$ is a partial order, your definition is probably correct, though there is another notion of dense subset of a partial order that is important in set theory: $B$ is dense in $A$ if for each $a\in A$ there is a $b\in B$ such that $b\preceq a$.