Understanding transversality of maps I'm trying to understand the definition of transversality of a map: A map $f:M \to N$ is transversal to a submanifold $A \subseteq N$ if whenever $f(x) = y \in A$ then $A_y + T_x f(M_x) = N_y$. It means the tangent space to $N$ at $y$ is spanned by the tangent space to $A$ at $y$ and the image of the tangent space to $M$ at $x$. 
Let $M=S^1$ and $N=S^2$. First I tried to work out what happens in the case $A$ is an open subset (ball) of $S^2$. Now I think that maybe it doesn't work at all because to have a tangent space to a set at a point $y$ would mean that $y$ lies in the boundary of the set. Is this correct? Next I thought $A$ could be a loop like $f(S^1)$. Then the tangent at $A$ seems to work but then I ran into the problem that both tangent lines $A_y $ and $ T_x f(M_x) $ appear to be parallel. What am I doing wrong here?
 A: If $A$ is an open subset of a manifold $N$, and $y \in A$, the tangent space $A_y$ is equal to $T_y N$. In particular it's not true that $y$ has to lie on the boundary of $A$ in order for $A_y$ to make sense (the tangent space $A_y$ makes sense when and only when $y \in A$; the particular definition depends on how you're defining the notion of "submanifold"). It follows that if $A$ is an open subset of $N$, then any map $f : M \to N$ is transversal to $A$.
Now the case $f : S^1 \to S^2$, with $A = f(S^1)$: Here $f$ certainly isn't transversal to $A$: as you say $A_y$ and $T_{f(x)} f(M_x)$ are identical (essentially by definition). In general it's rare for $f$ to be transversal to its image $f(M)$ (again by definition, this is equivalent to $f$ being a submersion).
As a third example you could let $f : S^1 \to S^2$ be the inclusion of the equator into $S^2$, and you could let $A$ be any great circle of $S^2$ other than the equator. $f(S^1)$ and $A$ intersect in exactly two points and the intersection is "transversal" at these points (meaning exactly your condition that the tangent spaces span). So in this case $f$ is tranversal to $A$.
