Yeah I have no idea what the author is getting at there. One obvious way to put a topology on the space of binary streams is to think of them as binary expansions of real numbers in the interval [0, 1] (the notation is supposed to indicate that we include 0 and 1) and inherit the topology from the normal topology on the reals in which case the answer is yes [0,1] is compact.

In this representation 0 is included as the binary stream consisting of all zeros and 1 is included as the binary stream consisting of all 1s.

This isn't entirely clean as a bunch of (rational) real numbers in [0,1] have multiple different binary expansions, for example 1/2 can be written

.10000... (the zeros go on forever)

or

.01111... (the ones go on forever)