Interiors of sets in topological vector spaces

[AxiomOfChoice]

In Rudin's book Functional Analysis, he makes the following claim about the interior $A^\circ$ of a subset $A$ of a topological vector space $X$: If $0 < |\alpha| \leq 1$ for $\alpha \in \mathbb C$, it follows that

$$\alpha A^\circ = (\alpha A)^\circ,$$

since scalar multiplicaiton (the mapping $f_\alpha: X \to X$ given by $f_\alpha(x) = \alpha x$) is a homeomorphism; that is, a 1-to-1 continuous mapping.

This is the first time I've ever confronted the term "homeomorphism," and I have no idea how I can deduce the above observation about interiors given this hypothesis. Can someone help? (This is not a HW problem; it's a point in our notes that is confusing me.)

 

[Landau]

You haven't seen the definition of homeomorphism? Its the topological version of 'isomorphism': an invertible continuous function with continuous inverse; this must be in Rudin also. Note: you say "a 1-to-1 continuous mapping", this is wrong. I hope by 1-1 you mean 'bijective' (I find this a confusing term, I have seen people use it for 'injective'), but the inverse is also required to be continuous!