# A topological problem in defining the tangent bundle

In one of the questions of my homework, tangent bundles are defined as in the picture. So I was wondering how to prove the openness of $$\pi^{-1}(U)$$, since the question only states that $$\pi^{-1}(U)$$ with the subspace topology is hemeomorphic to $$V\times \mathbb{R}^n$$. Actually, I do not understand how the topology is defined at all. (I do know how the tangent bundle is usually defined).

## 1 Answer

Well you see that $$T\varphi$$ is a bijection, so since you know the product topology on $$V \times \mathbb{R}^n$$, that automatically give you a topology on $$\pi^{-1}(U)$$. If you want, you give $$\pi^{-1}(U)$$ the topology which makes the map $$T\varphi$$ a homeomorphism.

This is the analogue of when you define the quotient topology, that is you give the target space the topology which makes the surjection continous.

• You mean that $\pi^{-1}(U)$ is open by definition? – user150248 Oct 8 '18 at 0:29
• Yes, but also any subset of $\pi^{-1}(U)$ is open iff its image uder $T\varphi$ is open. – Baol Oct 8 '18 at 13:43