  We construct quasi one-dimensional topological and non-topological three-band
lattices with tunable band gap and winding number of the flat band. Using mean
field (MF) and exact density matrix renormalization group (DMRG) calculations,
we show explicitly how the band gap affects pairing and superconductivity (SC)
in a Hubbard model with attractive interactions. We show excellent agreement
between MF and DMRG. When a phase twist is applied on the system, a phase
difference appears between pairing order parameters on different sublattices,
and this plays a very important role in the SC density. The SC weight, $D_s$,
on the gapped topological, $W\neq0$, flat band increases linearly with
interaction strength, $U$, for low values, and with a slope that depends on the
details of the compact localized state at $U=0$. As $U\to 0$ for the gapped
non-topological flat band ($W=0$), $D_s$ decays with a power law faster than
quadratic but slower than exponential. This indicates that isolated
non-topological flat bands are less beneficial to SC. In the gapless case (flat
band touching the band above it), we find at low $U$ (both for $W=0$ and $W\neq
0$) that $D_s\propto U^\varphi$ with $\varphi<1$ contrary to the $U{\rm ln}\,
({\rm const.}/U)$ behavior reported in the literature. In other words, $D_s$
increases faster than linearly for low $U$ thus favoring SC at weak interaction
more than the gapped case. For systems with touching bands, we observe that the
one-body correlation length, $\xi$, diverges as a power law as $U\rightarrow0$,
while for the isolated flat band $\xi(U\to 0)$ is a constant smaller than one
lattice spacing. Both behaviors are distinct from the exponential divergence of
$\xi$ in the dispersive case. Our results re-establish that the BCS mean field
and quantum metric alone are insufficient to characterize SC at weak coupling.
