Metric Space and Analysis: Prove that if $U$ is an open subset of $\mathbb R$ and $c\in U$, then $U\setminus\{c\}$ is disconnected

Question

Prove that if $U$ is an open subset of $\mathbb{R}$ and $c\in U$, then $U\setminus \{c\}$ is disconnected.

Answer 1

If you can contain your set into two open disjoint sets, then your set is disconnected.

Think about it.

Are there two disjoint open sets covering $R-{c}?

Answer 2

I offer the following generalization of this statement, which is intended to be a hint as to the direction that will lead you to the solution.

Suppose that $U$ is a (not necessarily open) subset of $\mathbb R$ and $c\in U$. Suppose, furthermore, that there exist real numbers $a,b\in U$ such that $a<c<b$.

Then, $U\setminus\{c\}$ is disconnected.

To-do list:

• Prove that the highlighted statement is true.
• Show that a non-empty open subset of $\mathbb R$ satisfies the premise of the above statement. Conclude.

Answer 3

Let $A = U\cap(-\infty,c), B = U\cap (c,\infty)$, then $A,B$ are both open w.r.t $\mathbb{R}$ and $U \subseteq A \cup B$, and further $A \cap B = \emptyset$. This shows $U \setminus \{c\}$ is disconnected.