The interior of an arbitrary union contains the union of the interiors

up:: Interior (Topology)

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

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

Since All sets contain their interior, we have that $\mathring{M_i}\subseteq M_i$.

Since $\bigcup _{i\in \Lambda }\mathring{M_i}\subseteq \bigcup _{i\in \Lambda }{M}_i$, and since Interior preserves subset ordering, we have that

$$\bigcup _{i\in \Lambda }\mathring{M_i}\subseteq {\left(\bigcup _{i\in \Lambda }{M}_i\right)}^{\circ }$$

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References

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