Question about connectedness of intervals on $ \mathbb{R} $

Question

I am starting to learn about connected topological spaces and one recurring fact is that any intervals $ [a, b] $ is connected in $ \mathbb{R} $. The proof of this fact can be found easily online, for example here and here. My question is, suppose I take $ \mathbb{R} $ with the discrete topology and let $ [a, b] $ be an interval. Then $ [a, b] $ is a subspace of $ \mathbb{R} $ with the discrete topology as well. For some $ c \in [a, b] $, I can decompose $ [a, b] $ as $ [a, c] \coprod (c, b] $ and both $ [a, c) $ and $ [c, b] $ are nonempty, disjoint, and open in $ [a, b] $. This means $ [a, b] $ is not connected in $ \mathbb{R} $. Am I misunderstanding anything here because this contradicts the fact that $ [a, b] $ is connected in $ \mathbb{R} $.

Answer 1

Actually its a game of what topology you are actually choosing. Yes it's not connected with discrete topology but the argument you are asking is for usual topology.

Answer 2

Note that [a,b] is not connected in discrete topology.