For a vector space and a topology on the space, this pair is called a topological vector space when is closed for all and both vector space operations are continuous. This being addition and scalar multiplication.
Defining maps by where its not hard to show that is in fact a homeomorphism. This gives us that the topology is uniquely defined local, around some point say because is open iff is open for all .
I think this is pretty neat and possibly the atypical situation where things locally define everything globally thus illustrating a guiding philosophy in modern mathematics.