Prove that $[0,2)∪\{2 +1/n:n ∈ ℕ\}$ is connected.

Question

Prove that the set $A$ $=$ $[0,2)$ $∪$ $\{2 +1/n:n \in \mathbb{N}\}$ with the standard metric is connected.

Assume that $A$ is not connected, meaning that it can be written as the union of two non-empty, disjoint, open sets $U$ and $V$.Since $U$ and $V$ are open, for each point $x ∈ U$, there exists an $ε_x > 0$ such that the ball $B(x, ε_x) = \{y \in A : d(x, y) < ε_x\}$ is contained in $U$. Similarly, for each point $y ∈ V$, there exists an $ε_y > 0$ such that the ball $B(y, ε_y) = \{z \in A : d(y, z) < ε_y\}$ is contained in $V$.

Since $A$ is the union of $[0,2)$ and $\{2 + 1/n : n \in \mathbb{N} \}$, and $U$ and $V$ are disjoint, we must have that either $[0,2) \subseteq U$ and $\{2 + 1/n : n \in \mathbb{N} \} \subseteq V$, or vice versa.

NOW HOW TO APPROACH...

Answer 1

The set $A$ is disconneted since $A=\left(A\cap\left(-\infty,\frac{11}4\right)\right)\cup\left(\{3\}\cap\left(\frac{11}4,\infty\right)\right)$, and both $\left(A\cap\left(-\infty,\frac{11}4\right)\right)$ and $\left(\{3\}\cap\left(\frac{11}4,\infty\right)\right)$ are non-empty open disjoint subsets of $A$.