Every half-plane $H=\{(x,y)\in\mathbb{R}^{2}\mid ax+by+c\geq 0\}$ is connected 
Prove that every half-plane $$H=\{(x,y)\in\mathbb{R}^{2}\mid ax+by+c\geq 0\},$$ where $a,b,c\in\mathbb{R}$, is connected.

Assume that $H$ is disconnected (or that there is a separation for it). We can write $H$ as union of two non-empty and disjoint subsets of $H$. Now,
$$H=U\cup V=W\cap(U\cup V)=(W\cap U)\cup (W\cap V), $$
where $W\in\mathbb{R}^{2}$ is open. We also know that $U,V$ are clopen, because we assumed that $H$ is disconnected.
Then I am stuck. I need a hint to get forward.
Update, check hints below. I'll rewrite this at better time.
 A: Hint Show that $H$ is starlike, that is, that for some fixed $(x_0, y_0)$, for every $(x, y) \in H$, the line segment with endpoints $(x_0, y_0)$ and $(x, y)$ is contained in $H$; in particular, this shows that $H$ is path-connected, and hence connected. (It is convenient here to take $(x_0, y_0)$ to lie on the boundary $\{a x + b y + c = 0\}$ of the space.)
A: Both Travis and Soma have good ideas. You may:


*

*Show that the half-plane is path-connected. To see this, pick any two points $x,y$ inside the half-plane, and show that the straight line $(1-t)x + ty$ connecting $x$ to $y$ belongs to the half-plane for every $t \in [0,1]$. (In fact this shows convexity.) And indeed we have 
$$a((1-t)x_1 + ty_1) + b((1-t)x_2 + ty_2) + c = (1-t)(ax_1+bx_2+c) + t(ay_1+by_2+c) \geq 0. $$

*Find a continuous map of $\mathbb{R}^2$ onto the half-plane. You may for instance consider the map which is the identity inside the half-plane and pushes points outside to the boundary along the normal vector $(a,b)$.
A: For another approach, you might work in $\mathbb C$ and observe that $z \mapsto \frac{1+z}{1-z}$ maps the open unit disc onto $\{z\in\mathbb{C}:\Re(z)>0\}$ and that $H$ can then be obtained by taking the closure and doing a rotation and translation. 
A: Piggybacking off of Travis' answer, there's a useful theorem about connectivity:
Let $\{U_i\}_{i \in I}$ be a family of connected sets such that all $U$ pairwise intersect (any two $U_j,U_k$ have nonempty intersection). Then $\bigcup_i U$ is connected. 
You can also use this instead of path connected $\Longrightarrow$ connected, along with writing $H$ as a bunch of line-segments that intersect at some point $(x_0,y_0)$. 
