I'm a undergrad student and getting started Topology. Today, when I check the continuity of a function in the indiscrete topology T = {phi, X}, several functions that are not continuous in the usual topology become continuous. Why does this happen?
Am I correct in saying that continuity depends on the choice of topology?
For example, f(x) = x\^2 is continuous everywhere, even in the indiscrete topology, which seems fine. But when I consider the Dirichlet function, f(x) = 1, if x is rational f(x) = 0, if x is irrational,
it also becomes continuous, which feels very counterintuitive.
If I have to choose the topology before checking continuity, then why do we even need the notion of continuity in topology in the first place? Doesn't the choice of topology already determine the answer?
I'm really confused about this. Could you please explain the intuition behind it?
Thank you for reading