I am trying to understand the nuances among different possibly correct definitions over some core concept in calculus/ real analysis to deepen my understanding. I attempted some true/false questions. Let me know if I am correct. NOTE: let me know if I am wrong, I will try to justify and pinpoint my confusion! Thanks a ton for people helped!
1. True, it's stardard defintion I learn so it's obvious
2.F. Because there is no 0<|x|
3.True. I feel like its true..
4.False Because 5 is true
5.T. Because the preimage of an open set is open