  Topological phenomena are commonly studied in phases of matter which are
separated from a trivial phase by an unavoidable quantum phase transition. This
can be overly restrictive, leaving out scenarios of practical relevance --
similar to the distinction between liquid water and vapor. Indeed, we show that
topological phenomena can be stable over a large part of parameter space even
when the bulk is strictly speaking in a trivial phase of matter. In particular,
we focus on symmetry-protected topological phases which can be trivialized by
extending the symmetry group. The topological Haldane phase in spin chains
serves as a paradigmatic example where the $SO(3)$ symmetry is extended to
$SU(2)$ by tuning away from the Mott limit. Although the Haldane phase is then
adiabatically connected to a product state, we show that characteristic
phenomena -- edge modes, entanglement degeneracies and bulk phase transitions
-- remain parametrically stable. This stability is due to a separation of
energy scales, characterized by quantized invariants which are well-defined
when a subgroup of the symmetry only acts on high-energy degrees of freedom.
The low-energy symmetry group is a quotient group whose emergent anomalies
stabilize edge modes and unnecessary criticality, which can occur in any
dimension.
