Let $p:E\to B$ be a smooth fiber bundle with fiber $F$. Consider the vector spaces $V_u=\{x\in T_uE: p_*(x)=0\}$. We call $V_u$ the vertical subspace of the tangent space $T_uE$.
How can we see that $V_u$ is the tangent space to $E_u$, the fiber over $u$? What is the intuition to call them vertical?
How does $T_uE_u$ sit in $T_uE$? Do we speak of $T_uE_u$'s that are close together? How does $T_uE_u$ sit against $T_vE_v$? Which are the transition maps for the subbundle $\cup_u T_uE_u$? Which is its topology?
Thought: If $E$ is a principle $F$-bundle, and the Lie algebra of $F$ is $\mathfrak{f}$, we can identify $V_u$ with $\mathfrak{f}$. In this situation we partially answered the first question. Despite having an identification I do not see that $V_u$ is tangent to $E_u$. Likely, the subset $V=\cup_u V_u$ is bundle called the vertical subbundle.
What is an example of a tangent bundle to a bundle that we can embed in three space allowing us to depict all its relevant subspace and projections?
Do any topological or geometrical considerations (possibly adding to the case of a principle $F$-bundle) lead us to say that $V_u$ is $T_uE_u$?