  Scanning tunneling microscopy/spectroscopy (STM/STS) measurements were
carried out on a multi-layered cuprate superconductor
Ba$_2$Ca$_5$Cu$_6$O$_{12}$(O$_{1-x}$,F$_x$)$_2$. STM topography revealed random
spot structures with the characteristic length $\le 0.5$ nm. The conductance
spectra dI/dV(V) show the coexistence of smaller gaps $\Delta_S$ and large gaps
(pseudogaps) $\Delta_L$. The pseudogap-related features in the superconducting
state were traced with the spatial resolution of $\sim$ 0.07 nm. Here, $I$ and
$V$ are the tunnel current and bias voltage, respectively. The temperature,
$T$, dependence of $\Delta_S$ follows the reduced Bardeen-Cooper-Schrieffer
(BCS) dependence. The hallmark ratio 2$\Delta_{S}(T=0)/k_B T_c$ equals to 4.9,
which is smaller than those of other cuprate superconductors. Here, $T_c$ is
the superconducting critical temperature and $k_B$ is the Boltzmann constant.
The larger gap $\Delta_L$ survives in the normal state and even increases with
$T$ above $T_c$. The $T$ dependences of the spatial distributions for both
relevant gaps ($\Delta$ map), as well as for each gap separately ($\Delta_S$
and $\Delta_L$) were obtained. From the histogram of $\Delta$ map, the averaged
gap values were found to be $\bar \Delta_S = \sim 24$ meV and $\bar \Delta_L =
\sim 79$ meV. The smaller gap $\Delta_S$ shows a spatially homogeneous
distribution while the larger gap $\Delta_L$ is quite inhomogeneous, indicating
that rather homogeneous superconductivity coexists with the patchy distributed
pseudogap. The spatial variation length $\xi_{\Delta_L}$ of $\Delta_L$
correlates with the scale of the topography spot structures, being
approximately 0.4 nm. This value is considerably smaller than the coherence
length of this class of superconductors, suggesting that $\Delta_L$ is strongly
affected by the disorder of the apical O/F.
