Topology induced on an interval $[a,b] \subset \mathbb{R}$ and connected set definition.

Question

This is a stupid question.

Am I right when I say that the interval $[a,b]$ is not an open set w.r.t. the topology of $\mathbb{R}$ but it is an open set if I consider $[a,b]$ as a topological subspace of $\mathbb{R}$?

The reason I'm asking is because of the definition of connected set in a topological space, which according to this link means that the set cannot be partitioned in two open sets in the subspace topology I think.

Answer 1

To allay confusion, I would word this much more carefully, with care taken to match the general wording of open subsets.

When a topological space $X$ is given (with its topology specified somehow), one can ask whether a subset $A \subset X$ is open in $X$.

So, for example, when the topological space $\mathbb R$ is given (with its standard topology), its subset $[a,b] \subset \mathbb R$ is not open in $\mathbb R$.

But when the topological space $[a,b]$ is given (with the subspace topology that it inherits from $\mathbb R$), its subset $[a,b] \subset [a,b]$ is open in $[a,b]$.

Answer 2

Every topological space is an open subset of itself. So, yes, $[a,b]$ is an open subset of $[a,b]$ if we see $[a,b]$ as a topological space of $\mathbb R$.