A的闭包+B的闭包包含于A+B的闭包

Theorem

Let $X$ be a topological vector space. If $A\subset X$ and $B\subset X$, then $\overline{A}+\overline{B}\subset \overline{A+B}$.

proof

1. Take $a\in \overline{A}$, $b\in \overline{B}$; Let $W$ be a neighborhood of $a+b$.

2. There are neighborhoods $W_1$ and $W_2$ of $a$ and $b$ s.t. $W_1+W_2\subset W$.

3. There exists $x\in A\cap W_1$ and $y\in B\cap W_2$, since $a\in \overline{A}$ and $b\in \overline{B}$.

4. Then $x+y\in A+B$ and $x+y\in W$.

5. Hence $(A+B)\cap W\ne \varnothing$.

6. Consequently, $a+b\in \overline{A+B}$

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