The interior of a finite intersection is the intersection of the interiors

up:: Interior (Topology)

Let $(X,\tau )$ be a Topological Space.
Proof by induction:

1. Let $M,N\subseteq X$. Note that, since All sets contain their interior, we have that $\mathring{M},\mathring{N}\subseteq M,N$, which implies that $\mathring{M}\cup \mathring{N}\subseteq M\cup N$
2. Let $\bigcup _{i=1}^n\mathring{M_i}\subseteq \bigcup _{i=1}^n{M}_i$. Then, as above, $\bigcup _{i=1}^n\mathring{M_i}\cup \mathring{M_{n+1}}\subseteq \bigcup _{i=1}^{n+1}{M}_i$

Counterexample of interior of infinite intersections

Let $M_i=[-\frac{1}{n}+1,1+\frac{1}{n}],n\ge 2$ (sets around $x=1$)

• $\mathring{M_i}=(-\frac{1}{n}+1,1+\frac{1}{n})$ whose infinite intersection is $1$
• $\bigcap _{n=2}^{\infty }{M}_i=1$, whose interior is $\varnothing$

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References

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