I'm acquainting myself with the world of Lie groups, and have seen that since there is a bijection between left invariant vector fields and a tangent vector at the identity. And thus one defines the Lie algebra of a Lie group as $\mathfrak{g}=T_{1_G}(G)$. However in the book I am reading they discuss the exponential map and say that it is continuous, but with no mention of the topology on $\mathfrak{g}$.
My question is given such a Lie algebra $\mathfrak{g}$ is the topology induced from the tangent bundle? Is it simply the topology on $\mathbb{R}^d$ ($d$ being the dimension of the Lie group)? Or is it the topology such that Lie algebra is a topological vector space and the Lie bracket is smooth?