How to determine if a set is connected?

Question

I am given that $A \setminus B$ is equal to the values of $A$ that are not found in $B$. Then I am given a bunch of different sets. My task is to determine if these sets are connected.

Now, I get the principle of connected domains and that each pair of points should be possible to join through a smooth curve in the domain. However, I have no idea on how I am supposed to calculate if a set is connected or not. Is it trial and error for different values? How do I interpret the values I get?

For example, this is one of the sets I am given $$\mathbb{R}^3 \setminus \{(x,y,z): z=4x+7y+7\}$$

How should approach this problem?

Answer 1

Hint:

$$\mathbb{R}^3 \setminus \{(x,y,z): z=4x+7y+7\} = \{(x,y,z): z>4x+7y+7\} \cup \{(x,y,z): z<4x+7y+7\}$$

so your space can be written as a union of two nonempty disjoint sets which are both open in $\mathbb{R}^3 \setminus \{(x,y,z): z=4x+7y+7\}$.

Conclude that it is not connected.

Answer 2

Let $\{(x,y,z): z=4x+7y+7\}$ and $M:= \mathbb R^3 \setminus P$.

Then $(0,0,0), (1,1,1) \in M$. Now show that there is no continuous mapping $f:[0,1] \to \mathbb R^3$ with

$f(0)=(0,0,0), f(1)=(1,1,1)$ and $f([0,1]) \subseteq M$.