I have some problems understanding the proof of the following lemma:
Lemma: Let $x \in X, \ \ \ U, W \in \mathcal{U}, \ \ \ \mathcal{T(U)}$ is the topology on $X$. If there exists $V \in \mathcal{U}$ such that $U \circ V \subset W$, then $U[x] \subset intW[x]$ and $\overline{U[x]} \subset W[x]$.
$\mathcal{U}$ is a uniform structure on $X$
$\mathcal{U} \subset 2^{X \times X}$ is a uniform structure on $X$. It is a filter on $X \times X$ satisfying:
$(1) \ \forall U \in \mathcal{U}: \Delta(X) \subset U, \\ (2) \ \forall U \in \mathcal{U}: \ U^{-1} \in U, \\(3) \ U \in \mathcal{U} \Rightarrow \exists V \in \mathcal{U} : \ V \circ V \subset U $.
$U[x] = \{ y \in X \ | \ (x,y) \in U\} $
If $\mathcal{B}$ is a basis of $\mathcal{U} \ \ \ \ $ ($ \ \forall U \in \mathcal{U} \ \exists B \in \mathcal{B} : \ B \subset U \ $),
then $ \mathcal{B_x}: = \{U[x] \ | \ U \in \mathcal{B}\}$ satisfies all conditions of basis for the neighbourhood system of $x \in X$ and $\mathcal{T(U)}$ mentioned above is a topology for which $ \mathcal\{{B_x}\}_{x \in X}$ is a basis for the neighbourhood system in this topology.
I hope my question is clear now.
Could you help me understand the proof of this lemma?
My problem is that I don't have a clear picture of what int$W[x]$ or $ \overline{U[x]}$ should be.
Here is the proof:
Let $y \in U[x], \ z \in V[y]$. Then $(x,z) \in U \circ V$, so $(x,z) \in W$. Thus $U[x] \subset intW[x]$.
Let $y \in \overline{U[x]}$. Then $\forall Z \in \mathcal{U} \ \ \exists x_Z \in U[x] \cap Z[y]$, which means that $(x, x_Z) \in U, \ (y, x_Z) \in Z$.
We choose $Z$ such that $Z = Z^{-1}$ and $Z \subset V$. Then $(x,y) \in U \circ Z \subset W$.
The proof given is some basic information about neighborhoods in a uniform space. Probably it's a hint.
well I prefer the normal notation $V\circ U$ for your $U\circ V$. The first part seems trivial: $$V\circ U\subseteq W \to$$ $$ V[U[x]]=(V\circ U)[x]\subseteq W[x] \to $$ $$\bigcup_{y\in U[x]}V[y]\subseteq W[x] \to$$ $$U[x]\subseteq W[x]^o $$
Last $\to$ is because $V[y]$ is a neighborhood of $y$ as said in the hint.
For the second part, I prefer to use nets.