# Simple topological problem

1. Sep 29, 2007

### r4nd0m

I'm really stuck on this simple problem: Let X be a topological vector space and U, V are open sets in X. Prove that U+V is open.

It should be a direct consequence of the continuity of addition in topological vector spaces. But continuity states that the $$f^{-1}(V)$$ is open whenever V is open, but not the converse. It would work if I showed that adding a constant is a homeomorphism, but I don't think this is the way I should do it. Is there any more simple way, that I overlooked?

2. Sep 29, 2007

### jostpuur

In fact it is sufficient to assume the other one of the sets to be open. Suppose V is open. If you succeed in proving that for all vectors u the set

$$u+V = \{u+v\;|\;v\in V\}$$

is open, then you are almost done.