Example 4 If three lines $a$, $b$, and $c$ in space are pairwise skew lines, then the number of lines that intersect with $a$, $b$, and $c$ is ( ).
(A) 0
(B) 2
(C) More than 1 but finite
(D) Infinite
(1997, National High School Mathematics Competition)
Regardless of the positional relationship between lines $a$, $b$, and $c$, a parallelepiped $A B C D-A_{1} B_{1} C_{1} D_{1}$ can always be constructed such that $A B$ lies on line $a$, $B_{1} C_{1}$ lies on line $b$, and $D_{1} D$ lies on line $c$, as shown in Figure 4.

Take a point $M$ on the extension of $D D_{1}$. Determine a plane $\alpha$ with point $M$ and line $a$. Plane $\alpha$ intersects lines $B_{1} C_{1}$ and $A_{1} D_{1}$ at points $P$ and $Q$, respectively. Then $P Q \parallel$ plane $\alpha$.
Thus, within plane $\alpha$, line $P M$ does not parallel line $a$,
so $P M$ must intersect line $a$ at a point $N$.
Therefore, line $M N$ intersects lines $a$, $b$, and $c$ simultaneously.
Since there are infinitely many ways to choose point $M$, there are infinitely many lines that intersect lines $a$, $b$, and $c$ simultaneously.