  The bulk-edge correspondence characterizes topological insulators and
superconductors. We generalize this concept to the bulk-corner correspondence
and the edge-corner correspondence in two dimensions. In the bulk-corner
(edge-corner) correspondence, the topological number is defined for the bulk
(edge), while the topological phase is evidenced by the emergence of
zero-energy corner states. It is shown that the boundary-obstructed topological
phases recently proposed are the edge-corner-correspondence type, while the
higher-order topological phases are classified into the
bulk-corner-correspondence type and the edge-corner-correspondence type. We
construct a simple model exhibiting the edge-corner correspondence based on two
Chern insulators having the $s$-wave, $d$-wave and $s_{\pm }$-wave pairings. It
is possible to define topological numbers for the edge Hamiltonians, and we
have zero-energy corner states in the topological phase. The emergence of
zero-energy corner states is observable by measuring the impedance resonance in
topological electric circuits.
