How to prove that Sum of two open sets is open?

Question

Let $X$ and $Y$ be two subsets of $\Bbb R$. Define $X + Y = \{x + y : x \in X, y \in Y\}$ How to prove that $X + Y$ is open ?

I see somewhere that $X + Y$ is open if either $X$ or $Y$ or both $X$ and $Y$ are open.

(1) I know that $X$ is open if $X^c$ is closed. Where $X^c$ denotes the complement of $X$

(2) $X$ is called open if it is neighborhood of each of its points.

(3) $X \cup Y$ and $X \cap Y$ are open if $X$ and $Y$ are open.

Say $X$ is open. Then $X+Y=\bigcup_{y\in Y}(X+y)$ is a union of open sets. — Ningxin, Jul 09 '21 at 20:01

Philipp · Accepted Answer · 2021-07-09T20:14:41.077

2

A subset of $\mathbb{R}$ is open if each of its points is contained in an open interval that lies in the subset.

Let $x+y \in X + Y$. By openness of $Y$, there is an open interval $I_y$ with $y \in I_y \subset Y$. Now, $x + I_y \subset X + Y$ is also an open interval that contains $x+y$.

Note that we have only used openness of one of the sets.

edited Jul 09 '21 at 20:14

answered Jul 09 '21 at 20:00

Philipp

169

How $x + I_y$ is open ? – Sangam Academy Jul 09 '21 at 20:04
2

$I_y$ is of the form $(a,b)$ for some $a,b \in \mathbb{R}$. Now, $x+I_y = (x+a, x+b)$ which is also an open interval. – Philipp Jul 09 '21 at 20:07
It means $X + Y$ is open when $Y$ is open and $X$ is either open or closed. – Sangam Academy Jul 10 '21 at 02:02
1

The sum of two sets (in $\mathbb{R}$ with standard topology) is open if one of them is open. No assumption on the other set is needed. – Philipp Jul 10 '21 at 08:17

How to prove that Sum of two open sets is open?

1 Answers1