The closure of an arbitrary intersection is contained in the intersection of the closures

up:: Closure (Topology)

Let $(X,\tau )$ be a Topological Space.

Let $\{M_i{\}}_{i\in \Lambda }\in \mathbb{P}(X)$.

Since All sets are contained inside their closure, we have that $M_i\subseteq \overline{M_i}$.

Since $\bigcap _{i\in \Lambda }{M}_i\subseteq \bigcap _{i\in \Lambda }\overline{M_i}$, and since Closure preserves subset ordering, we have that

$$\overline{\bigcap _{i\in \Lambda }{M}_i}\subseteq \bigcap _{i\in \Lambda }\overline{M_i}$$

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References

• Notas para um Curso de Física-Matemática, João Carlos Alves Barata. Cap. 27