Is the sum of an open set and any set always open in R²?

[Mr_Physics]

Homework Statement

Let X and Y be subsets of R^2, both non-empty. If X is open, the the sum X+Y is open.

This is either supposed to be proved or disproved.

Homework Equations

The Attempt at a Solution

This strikes me as false since we are only given the X is open. However, I'm not sure how to disprove it other than creating a direct counter example. Any thoughts?

 

[jbunniii]

What can you say about X + {y} if {y} is a singleton? Can you generalize from this?

 

[Mr_Physics]

No...I think I might need a hint.

 

[jbunniii]

X + {y} is just a translated copy of X. Specifically, if y = (a,b), then X + {y} is just X shifted right by a and up by b.

So if X is open, then X + {y} is open.

Now, think about X + Y, where Y is any nonempty set. Think of Y as a union of singletons {y}. Can you express X + Y in terms of the sets X + {y} where {y} are the singletons contained in Y?