I refer to the book http://www.math.colostate.edu/~renzo/teaching/Toric14/CoxLittleShenck.pdf.
Let $X_{\Sigma}$ be the toric variety of fan $\Sigma$. Let $\sigma$ be a cone in $\Sigma$. We define $O(\sigma)$ as the orbit of the distinguished point $\gamma_{\sigma} \in U_{\sigma}$ that is $O(\sigma) = T_N \cdot \gamma_{\sigma} \subseteq X_{\Sigma}$. Moreover one can prove that $O(\sigma) = \{\gamma : S_{\sigma} \to \mathbb{C} \; | \; \gamma(m) \neq 0 \leftrightarrow m \in \sigma^{\bot} \cap M\}$ and that $O(\sigma)$ is invariant with respect to the torus action, that is, $T_N \times O(\sigma) \to O(\sigma)$.
I have to prove that the closure of $O(\sigma)$ in the classical topology is invariant with respect to the torus action but I don't know how to proceed. Any suggestion?
This is true more generally -
Let $G$ be a topological group acting continuously on a topological space $X$. Let $A$ be a $G$ - invariant subspace of $X$. Then $\bar A$ is also $G$ - invariant
Proof :
Let $g\in G$ and $x\in \bar A$. To prove that $g\cdot x \in \bar A$
Since $x\in \bar A$, there exists a net of points $\{a_\alpha\}\subseteq A$ such that $a_\alpha\longrightarrow x$ . Since $G$ acts continuously, $g\cdot a_\alpha\longrightarrow g\cdot x$ . Then since $A$ is $G$ - invariant, $\{g\cdot a_\alpha\}$ is a net of points in $A$ converging to $g\cdot x$ $\Longrightarrow g\cdot x\in \bar A$. $\square$