Compatibility summary
You will need a travel adapter in Christmas Island; plug types used there differ from South Korea.
Plug comparison
South Korea plug types: C F
Christmas Island plug types: I
There is no common plug type between South Korea (C F) and Christmas Island (I). You need a travel adapter.
Voltage comparison
South Korea: 220V, 60Hz
Christmas Island: 230V, 50Hz
Voltage difference is within a range many devices tolerate; check your appliance label.