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**"Adding" 2 open sets**

## Homework Statement

I'm trying to prove that If both S and T are open sets then S+T is open set as well.

## Homework Equations

[itex]S+T=\{s+t \| s \in S, t \in T\}[/itex]

## The Attempt at a Solution

S+T is open if every point [itex] x_0 \in S+T [/itex] is inner point.

Let [itex]x_0[/itex] be a point in S+T, so there is [itex]s_0[/itex] in S and [itex]t_0[/itex] in T so that [itex]x_0=s_0+t_0[/itex].

S is open so for every ||s-s_0|| < δ_1 s in S.

T is open so for every ||t-t_0|| < δ_2 t in T.

Let x be point in S([itex]x_0[/itex], _delta_), I will write x=s+t. [both s and t are some vectors in R^n]

s+t in S([itex]x_0[/itex], _delta_)={s+t | ||[itex]s+t-s_0-t_0[/itex]|| < _delta_} and here I stuck, if I could conclude from ||[itex]s+t-s_0-t_0[/itex]|| < _delta_ that ||[itex]s-s_0[/itex]|| < δ_1 and ||[itex]t-t_0[/itex]|| < δ_2 the proof will be over, however I just can't find the algebraic manipulation.

Will appreciate any help.

Thanks.

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