Can someone please explain to me the idea of connectedness and how they relate to these sets? I understand the definition of connectedness, but I am struggling to relate the definition to these examples. I am faced with the following problem:
Show that each of the following sets are connected:
a) The grid $$G = \{x \in \mathbb{R}^2: x = (x_1, x_2) \text{ and at least one } x_i \in \mathbb{Z}\}\text{.}$$
b) The cone $$K = \{x \in \mathbb{R}^{d+1}: x = (s, y) \text{ where } y \in \mathbb{R}^d \text{ and } |s| = ||y|| \}\text{.}$$
c) Any convex set $C \subseteq R^d$. (Recall that a set $C$ is convex iff for each $x, y \in C$ the line segment $\{\lambda x+(1−\lambda)y: \lambda \in [0,1]\} \subseteq C$.