Checking continuity of stereographic projection in two ways Consider the topological spaces $(S^2\smallsetminus{N},\tau_a)$, where $N=(0,0,1)$ and $\tau_a$ is the subspace topology induced by the standard topology $\tau_{\mathbb{R^3}}$ on $\mathbb{R}^3$, and $(\mathbb R^2,\tau_{\mathbb{R}^2})$. Then consider the stereographic projection $S^2\smallsetminus{N}\to$ and its inverse.
The definition of continuity I have seen is that preimages of open sets are open. In this specific example I can also say that stereographic projection maps circles to circles.
But there is another way I can see continuity: if each component of the stereographic projection and its inverse are "visibly" compositions of continuous functions.
My question is: why is the second approach right? If the functions were functions of $\mathbb {R^3}$ I would see no problem, but in this case I have two different topologies on two different spaces and I should use the open set definition.
How can I switch from one definition to another without problems?
 A: I'd rather make this a comment, but I don't have enough reputation. Your question is a fair one, you are just missing the concept of product topology, which can be a bit confusing at first but is very important (and a good introduction to universal properties). First, at least if you are studying general topology, you should consider the preimages-of-open-sets-are-open definition of continuity as your base definition, every other definition of continuity has to be a special case of this one. In particular, you might have encountered other definition of continuity studying analysis, but one can prove that they coincide with the "true" definition of continuity (it is only with more general spaces than the ones you encounter in basic analysis that the "true" definition becomes indispensable). Now, the reason why in analysis we can check continuity of a function $X\to\mathbb{R}^n$ on its components is that the topology on the product space $\mathbb{R}^n$ is defined as the unique topology for which the continuous maps with target $\mathbb{R}^n$ are precisely those which are given by continuous maps to each of the factors of the product: thus giving a continuous map, $X\to\mathbb{R}^n$ is exactly the same as giving $n$ continuous maps $X\to\mathbb{R}$. This can be generalized to products of more general topological spaces than $\mathbb{R}$, which is the content of the Wikipedia article I linked, and you can find it in most books which cover general topology (say Topology by Munkres or, assuming by your name that you are Italian, Geometria 2 by Sernesi). As for the inverse map, if you understand the subspace topology, you will see that you can make a very similar argument (also, you don't need that circles are sent to circles, which is not really true anyway, only some of them are).
