# 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 $B \subseteq A \,\land\, A \neq B$.

-
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 $b\preceq a$.

-
M.Scott, un linear order is not a total order, is correct? – mle Aug 19 '13 at 20:37
@Soviet: Linear order is another term for total order; they mean the same thing. I’m more accustomed to the term linear order, so I usually use it. – Brian M. Scott Aug 19 '13 at 20:41
@Soviet: For some reason they choose to limit the term linear order to what I would call a strict linear order; they use total order for what I would call simply a linear order or, if there were any possibility of confusion, a non-strict linear order. – Brian M. Scott Aug 19 '13 at 20:47
okok I understand....thanks soo much!! – mle Aug 19 '13 at 20:48

According to Wikipedia, your definition is correct.

-