Topology: An Inquiry-Based Approach

With Theorem 18.5 in hand, we are just about ready to show that any interval in $\mathbb{R}$ is connected. Let us return for a moment to our example of $A=(-1,0)\cup (4,5)$ in $\mathbb{R}\text{.}$ It is not difficult to see that if $U$ and $V$ are a separation of $A\text{,}$ then the subset $(-1,0)$ must be entirely contained in either $U$ or in $V\text{.}$ The reason for this is that $(-1,0)$ is a connected subset of $A\text{.}$ This result is true in general.

Now we can prove that any interval in $\mathbb{R}$ is connected. Since $[a,b]\text{,}$ $[a,b)\text{,}$ and $(a,b]$ are all sets that lie between $(a,b)$ and $\overline{(a,b)}\text{,}$ we can address their connectedness all at once with the next result.

One consequence of Theorem 18.8 is that any interval of the form $[a,b)\text{,}$ $(a,b]\text{,}$ $[a,b]\text{,}$ $(-\mathrm{\infty },b]\text{,}$ or $[a,\mathrm{\infty })$ in $\mathbb{R}$ is connected. This prompts the question, are there any other subsets of $\mathbb{R}$ that are connected?