Openness of the chronological future and past, openness of the chronological relation, and closure of future and past sets

(Openness of $I^{+}(x)$.) Let $y\in I^{+}(x)$ and let $\gamma :[0,1]\to M$ be a future timelike curve from $x$ to $y$. Choose a strictly convex normal ball $B$ about $y$. By continuity, choose a fixed $t_0\in (0,1)$ sufficiently near $1$ that $\gamma ([t_0,1])\subseteq B$, and put $p_0:=\gamma (t_0)$. The tail is a future timelike curve in $B$, so $y\in I_B^{+}(p_0)$. By [L1], $I_B^{+}(p_0)$ is open in $M$; choose a neighborhood $V$ of $y$ contained in it. For every $z\in V$, we have $x\ll p_0$ by [L2] and $p_0\ll z$ by the definition of $I_B^{+}(p_0)$; transitivity [L3] gives $x\ll z$. Thus $V\subseteq I^{+}(x)$ and $I^{+}(x)$ is open. The time-reversed argument gives openness of $I^{-}(x)$. The center $p_0$ is fixed throughout the neighborhood argument; continuity is used only to place the original curve tail in the ball around $y$.

(Openness of the chronological relation.) Let $x\ll y$ and choose a future timelike curve $\gamma$ from $x$ to $y$; pick $t_0\in (0,1)$ and put $z:=\gamma (t_0)$, so that $x\ll z\ll y$ by [L2] and [L3]. By step 1.1 the sets $I^{-}(z)=\{a:a\ll z\}$ and $I^{+}(z)$ are open, and $I^{-}(z)\times I^{+}(z)$ is an open neighborhood of $(x,y)$ in $M\times M$; every $(a,b)$ in it satisfies $a\ll z\ll b$, hence $a\ll b$ by [L3]. This proves (b).

(Future sets are closed under the chronological future of their closure.) Let $F$ be a future set and $q\in \bar{F}$. Let $r\in I^{+}(q)$. By step 2.1, the set $I^{-}(r)$ is an open neighborhood of $q$, so it meets $F$; choose one $f\in F\cap I^{-}(r)$. Then $f\ll r$, and $r\in I^{+}(f)\subseteq I^{+}(F)\subseteq F$ by [L4]. Hence $I^{+}(q)\subseteq F$; the past statement is the time reverse. No sequence of chosen points is needed.

(Conclusion.) Step 1.1 proves (a), step 2.1 proves (b) and step 3.1 proves (c).