separability and connectness

Question

i have some confusion about real numbers and i will apreciate if someone can explain it to me.

So, we know that the set of real numbers is connected because there are no disjoint open sets that there union form the set of reals but since it has a countable dense subset ( set of rational numbers), so it is also a separable set !!! Can this be possbile !! Can a set be connected and separable in the same time ?!!

Answer 1

"Separable" does not mean "disconnected" (although in English the two words have similar meaning).

Separable means that the space has a countable dense set, nothing more.

So, yes, a space can be both separable and connected. The two have nothing to do with each other.

Answer 2

I think I understand the confusion since I was in the same boat once. These are two different notions: A separated space and a separable space. A separated space is one which is not connected. But a separable space is one that has a countable dense subset. Now as an example, if you take $\mathbb{R}$ and remove one point, the space becomes separated and separable. $\mathbb{R}$ by itself is connected and separable (set of rationals).

I'd like to add the reason separable is called as such is because it tells you that the essential properties(such as limit points, etc.) of the whole space (which is usually uncountable) may be understood by looking at a countable (and hence smaller) subset of it.