We study the effects of instantaneous feedback of channel topology on the degrees of freedom (DoF) of the bursty MIMO X channel, where the four transmitter-receiver links are intermittently on-and-off, governed by four independent Bernoulli $(p)$ random sequences, and each transmitter and receiver are equipped with $M$ and $N$ antennas, respectively. We partially characterize this channel: The sum DoF is characterized when $p\le \frac{1}{2}$ or when $\frac{\min(M,N)}{\max(M,N)} \le \frac{2}{3}$. In the remaining regime, the lower bound is within $5.2\%$ of the upper bound. Strictly higher DoF is achieved by coding across channel topologies. In particular, codes over as many as $5$ topologies are proposed to achieve the sum DoF of the channel when $p\le \frac{1}{2}$. A transfer function view of the network is employed to simplify the code design and to elucidate the fact that these are space-time codes, obtained by interference alignment over space and time.