On the Borel $\sigma$-algebra of a closed bounded interval in $\mathbb{R}$

Question

When we speak of the $\sigma$-algebra of the Borel sets in the bounded closed interval $[0,1]\subset \mathbb{R}$, we should consider the interval $[0,1]$ as a topological space. My doubt at this point is the following: When it is not mentioned any specific topology on $[0,1]$, do we consider $[0,1]$ as the topological subspace of $(\mathbb{R}, d_{\mathbb{R}})$, where $d_{\mathbb{R}}$ is the usual Euclidean metric on $\mathbb{R}$, or, do we consider $[0,1]$ as a topological space $([0,1], d_{\mathbb{R}}|_{[0,1]\times [0,1]})$, where $d_{\mathbb{R}}|_{[0,1]\times [0,1]}$ is the restriction of $d_{\mathbb{R}}$ to $[0,1]\times [0,1]$. The confusion arises for me, for example, if $\alpha\in (0,1)$ then, while the interval $[0,\alpha)=(-1,\alpha)\bigcap[0,1]$ is an open subset of $[0,1]$ with respect to the subspace topology on $[0,1]$, but it is not open in $([0,1], d_{\mathbb{R}}|_{[0,1]\times [0,1]})$. Am I right? Or, are these two topologies equivalent on $[0,1]$? Well, I am so confused!

Answer 1

For any set $A \subset \mathbb R$ the topology generated by the restriction of the usual metric to $A$ and the subspace topology of $A$ (consisting of the sets $A \cap U$ where $U$ is open in $\mathbb R$) are one and the same.