Upper semicontinuous function on a tangent bundle Let $\Omega$$\subset\mathbb{C}^n$ be a open connected set (domain). Let $T(\Omega)$ denote the tangent bundle of $\Omega$. Let $f:T(\Omega)\longrightarrow [0,\infty)$ be a map on $T(\Omega)$. 
What is the definition of $f$ bieng upper semicontinuous? 
The elements of $T(\Omega)$ look like $(z:\xi)$, where $\xi$ is a tangent vector at $z$. My intuition is: given any $\epsilon>0$ there exists a neighbourhood $U$ of $(z:\xi)$ such that $f(w:d)\geq f((z:\xi))+ \epsilon$ for every $(w:d)\in U$. But what is the topology on $T(\Omega)$ and what does a neighbourhood look like?
For example if we consider the Poincare metric on the unit disc $\mathbb{D}$ as
$\gamma(z:\xi)=\frac{|\xi|}{1–{|z|}^2}$. Then what is the definition of $ \gamma$ being upper semicontinuous? Also is there a sequential criteria?
 A: Following the wikipedia we say $f: X \to \mathbb R$ is upper semi-continuous, where $X$ is a topological space, if $f^{-1}(-\infty,y)$ is open for all $y \in \mathbb R$.
In your case we have $X=TM$. The tangent bundle is a manifold and therefore a topological space. More precisely. For each chart $\phi:U \to\mathbb R^n$, where $U$ is open subset of $M$, we have the vector fields $\partial/\partial x_j$ defined at $U$. The chart $\phi$ provides us the bijection $\Phi: \mathbb R^n \times \mathbb R^n \to TU$ defined by $$\Phi(x,v) = \sum_j v_j (\partial/\partial x_j)_\phi^{-1}(x).$$ The topology defined in $TM$ is the following: $A \subset TM$ is open if for each chart $\phi:U \to \mathbb R^n$ the subset $\Phi^{-1}(A \cap TU) \subset \mathbb R^n \times \mathbb R^n$ is open set. So, we have a topology on $TM$ and we can talk of semi-continuity. With this topology and with the maps $\Phi$ as charts we obtain $TM$ is an manifold.
In your example, we have $X= \mathbb D$, $TX = \mathbb D \times \mathbb R^2$ and $\gamma$ is continuous. Therefore upper semi-continuous.
There is also the concept of sequentially upper semi-continuous. We say $f:X \to \mathbb R$ is sequentially upper semi-continuous if for all convergent sequence $x_n \to x$ we have $\limsup f(x_n) \leq f(x)$. We have upper semi-continuity = sequentially upper semi-continuity if $X$ is first countable. Metric spaces and manifolds are first countable, so the two definitions are equivalent in these cases.
