nooo it is not
The most populous neighbour of Australia is Indonesia.
Australia does not own Antarctica: it is owned by no country.
There are no countries in Antarctica, and Australia is a country itself.
The continent of Antarctica is wholly located in the Southern Hemisphere. Nearby, you can find Chile, Argentina, New Zealand, Australia, South Africa and India.
Neighboring countries include New Zealand, Australia, Chile, Argentina, South Africa and India -- all separated from Antarctica by the Southern Ocean.
Antarctica is governed by the Antarctic Treaty: there are no countries there, as you state. Australia is a single country.
Australia is a country (and also the only country that is also a continent.) Antartica, however, is not.
Australia is in the Pacfics with neighbour New Zealands Singapore is in South-East Asia with neighbouring countries Indonesia and Malaysia.
There is no country "under" Australia. If you head south from Australia, you will come to Antarctica, which is a continent under no country's jurisdiction.
Australia and Antarctica.
Australia is a single country. There are no countries on the Antarctic continent.
Antarctica is a continent. It has no countries bordering it, it is completely surrounded by water.