why $\mathbb{R}$ is connected in usual Topology?

Question

I have found the link : Prove $\mathbb{R}$ is connected

But could not get the logics in my head

why $\mathbb{R}$ is connected in usual Topology ?

My thinking : I know that $\mathbb{R}$ is Hausdorff, then $\mathbb{R}$ must be disconnected

Answer 1

Hausdorff means that any two distinct points are contained in disjoint open sets. Connected means that the entire space can’t be partitioned into two nontrivial disjoint open sets. These are very different things, one very local in nature and the other very global. Hausdorff certainly does not imply disconnected.

The reals are connected in the usual topology because they satisfy the definition: you can’t partition the reals into nontrivial disjoint open sets. As a sketch of a proof, consider two points $a<b$ in the two different pieces $A$ and $B$. The least upper bound of $\{x: [a,x)\subset A\}$ is a boundary point in $A$ or $B.$

Answer 2

$\mathbb{R}$ is connected, because $\emptyset$ and $\mathbb{R}$ are the only sets which are clopen (closed and open) with regards to the standard topology on $\mathbb{R}$.

Answer 3

The real line is connected because there are "enough" real numbers to make it so, in a sense. I consider this a fairly nontrivial fact. Maybe read up on "Dedekind cuts", for instance.