4. Given a regular quadrilateral pyramid $S-A B C D$ with a height of 2 and a base edge length of $\sqrt{2}$, points $P, Q$ lie on segments $B D, S C$ respectively. The shortest distance between $P$ and $Q$ is
A. 1
B. $\frac{2 \sqrt{3}}{3}$
C. $\frac{2 \sqrt{5}}{5}$
D. $\frac{\sqrt{10}}{5}$
4. The shortest distance between $\mathrm{C} P, Q$ is the distance between the skew lines $B D$ and $S C$. Let the center of the square base $A B C D$ be $O$, then $B D \perp$ plane $S O C$.

Draw $O M \perp S C$ at $M$ from $O$, then $B D \perp O M$, thus $O M$ is the required distance.
Since $S O=2, O C=1$, hence $S C=\sqrt{5}$.
From $S O \times O C=S C \times O M$, we get $O M=\frac{2 \sqrt{5}}{5}$.