Some questions about open and closed maps Let $f:\mathbb{R}^2\rightarrow\mathbb{R}$ be a map as follows:
1.$f(x,y)=x+y$ is an open map or a closed map?
2.$f(x,y)=x^2-y^2$ is an open map or a closed map?
3.$f(x,y)=x^2+xy+y^2$ is an open map or a closed map?
4.$f(x,y)=(x+y)^2$ is an open map or a closed map?
5.$f(x,y)=x^3+y^3$ is an open map or a closed map?
6.$f(x,y)=x^3+y$ is an open map or a closed map?
7.$f(x,y)=x^5+y^2$ is an open map or a closed map?
Is there any (effective) necessary and sufficient condition to testify that a map is a closed map? 
Thanks a lot.
 A: Closedness
The first map is not closed since the closed set $((-n,n+1/n))_{n∈\Bbb N}$ has an image with accumulation point $0,$ which is not in the that image.
The same closed set is sent to a non-closed set by the fourth map.
The same idea can be used for the fifth map. Here the set $((-n,n+1/n^3))_{n\in\Bbb N}$ is mapped to a non-closed set.
Similar constructions can be used for the sixth and the seventh map.
The second map isn't closed either. Since the map is the composite $\mu\circ\begin{pmatrix}1 & -1 \\ 1 & 1\end{pmatrix}$, where $\mu:(x,y)\mapsto xy$, is suffices to study $\mu$. Now $((n,1/n^2))_n$ has the non-closed image $(1/n)_n$.
The third map, however, is closed. You can use that any proper map to a compactly generated Hausdorff space is a closed map. The third map $f$ is proper as $f(x,y)\to\infty$ for $\lVert (x,y)\rVert\to∞$.
Now on to openness
The first function is open: If $U$ is an open set in the plane and $(x,y)∈U$, then $f(U)$ contains the neighborhood $f(X×\{y\}∩U)$ of $x+y$. The same reasoning applies to map 5, 6, and 7.
The third and fourth map are not open as their range is $[0,\infty)$.
The second map is open. Can you prove it?
