  In this work, we reported a ubiquitous presence of topological Floquet time
crystal (TFTC) in one-dimensional periodically-driven systems. The rigidity and
realization of spontaneous discrete time-translation symmetry (DTS) breaking in
our model require necessarily coexistence of anomalous topological invariants
(0 modes and $\pi$ modes), instead of the presence of disorders or many-body
localization. We found that in a particular frequency range of the underlying
drive, the anomalous Floquet phase coexistence between zero and pi modes can
produce the period-doubling (2T, two cycles of the drive) that breaks the
spontaneously, leading to the subharmonic response ($\omega/2$, half the drive
frequency). The rigid period-oscillation is topologically-protected against
perturbations due to both non-trivially opening of 0 and $\pi$-gaps in the
quasienergy spectrum, thus, as a result, can be viewed as a specific "Rabi
oscillation" between two Floquet eigenstates with certain quasienergy splitting
$\pi/T$. Our modeling of the time-crystalline 'ground state' can be easily
realized in experimental platforms such as topological photonics and ultracold
fields. Also, our work can bring significant interests to explore topological
phase transition in Floquet systems and to bridge the gap between Floquet
topological insulators and photonics, and period-doubled time crystals.
