I cannot think of a word that means to deliberately not care. I thought ambivalent might fit; however, that means having conflicting feelings about a person or subject. I am searching for a word that means to not care either way something happens, almost as if this person thinks so lowly of another (perhaps considering them obsolete), they do not subject themselves to any involvement in that persons life. Any help is appreciated.
That person could be described as callous, and they would exhibit callous indifference.
Callous (MW, definition 2)
feeling no emotion
feeling or showing no sympathy for others
Plenty of examples at that link. Plus you could say:
Mary exhibited callous indifference when she didn't hold the door open for the man with a walker.
My boss is so callous - the first thing he said when I told him that I had the flu was "I hope you didn't infect anyone else."
See if this one fits the bill .......
showing or feeling a strong aversion. "it is human nature to be antipathetic to change"
synonyms: hostile to, against, (dead) set against, opposed to, antagonistic to/toward, ill-disposed to, unsympathetic to/toward; More informalanti, down on
Usage; "we were taught to be antipathetic to all foreigners"