Q: For the future Cauchy development (or future domain of dependence), why is $D^+ (S) \subset \tilde{D}^+ (S)$? Let $S \subset \mathcal{M}$ (where $\mathcal{M}$ is our manifold), the future Cauchy development of $S$ is defined as
$$D^+ (S) := \{p \in \mathcal{M} |\ \text{every past inextendible causal curve through} \ p \ \text{intersects} \ S \}$$
analogously, the same name is used for the following set
$$\tilde{D}^+ (S) := \{p \in \mathcal{M} |\ \text{every past inextendible timelike curve through} \ p \ \text{intersects} \ S \}$$
Both Minguzzi (2019) and Hawking & Ellis (1973) (in Remark 3.2 and Proposition 6.5.1 respectively) state that $D^+ (S) \subset \tilde{D}^+ (S)$. Why is this true? Both sources state it as a direct consequence from the former definitions with no further explanation, but I don't see it. A causal curve is either a null curve or a timelike curve. This means that for $p \in D^+ (S)$, there are more curves that pass through $p$ that intersect $S$ compared to the points in $\tilde{D}^+ (S)$, shouldn't this mean that $\tilde{D}^+ (S) \subset D^+ (S)$? After all, the timelike curves mentioned in the definition of $\tilde{D}^+ (S)$ are included in the definition of $D^+ (S)$.

A: Suppose there exists a point $p\in\mathcal{M}$ for which every past inextendible timelike curve through $p$ intersects $S$, but there exist an (inextendible) null curve through $p$ that does not intersect $S$. Then $p \in \tilde{D}^{+}(S)$, but $p \notin D^{+}(S)$.
