The closure of a finite union is the union of the closures

up:: Closure (Topology)

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

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

Counterexample of closure of infinite unions

Let $M_i=(-1-\frac{1}{n},1+\frac{1}{n}),n\ge 2$.

• $\overline{M_i}=\left[-1-\frac{1}{n},1+\frac{1}{n}\right]$ whose infinite union is $(-1,1)$
• $\bigcup _{n=2}^{\infty }{M}_i=(-1,1)$, whose closure is $[-1,1]$

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References

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