DB said:
A white dwarf about the size of the Earth but with a mass of a star leads to a very high density. An atomic structure where the electrons are packed very close the the nucleus, otherwise know as degenerate matter. Correct me if I'm wrong; this is due to cold temperatures...
DB the high density is not due to extreme cold, but rather to gravity
what keeps our sun from contracting to small size is the temperature at the core which is about 15 million kelvin.
(much more than the roughly 5000 kelvin at the surface!)
something 15 million kelvin glows Xrays, not ordinary light
the core MUST be this hot to create outwards pressure to fight against gravity.
when a star exhausts all the fuel, can no longer fuse nuclei, can no longer sustain the high (millions kelvin) core temp, then its core begins to cool and it begins to contract
(along the way there can be episodes where it can fuse other elements and temporarily re-expand, so it may not be a straight line decline)
You are right to study the HR diagram, the story of many stars is shown there.
The star can contract and become dwarf-dense even tho it is very hot!
It can be white hot. It can be 100s of thousands kelvin!
It can still contract, even tho very hot, because in its core it does not have the 10 million kelvin (or so, more or less) needed to keep in an expanded state.
so white dwarves can be hot and stay hot for a long time (gradually cooling as they radiate away their high temperature heat)
because they are so compact, have little surface area compared with mass, they cool slowly
probably someone will supply a link that says all this better. I have to go so no time to get a link