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Topological Spaces

In general relativity, spacetime has the same structure as a topological space. A topological space is a set $X$ equipped with a collection of open subsets $T$ that satisfy the following properties for $O\in T$:

1. The union of any collection of sets in $T$ is also in $T$: $\text{If } O_\alpha \in T \text{ for all } \alpha \in A, \text{ then } \bigcup_{\alpha \in A} O_\alpha \in T$, then:$$\bigcup_{\alpha \in A} O_\alpha \in T$$
2. The intersection of any finite number of sets in $T$ is also in $T$: $\text{If } O_1, O_2, \ldots, O_n \in T, \text{ then } U_1 \cap U_2 \cap \ldots \cap U_n \in T$, then:$$\bigcap_{i=1}^{n} O_i \in T$$
3. The empty set $\emptyset$ and the entire set $X$ are in $T$.