Is a set that consists of a single point connected?

Question

If $S=\{a\}$ then surely for every two points from that set there is a path that joins them, choose $x_1=a$ and $x_2=a$ and define continuous function on $[0,1]$ such that $f(x)=a$. Or, one point cannot be represented as union of two or more disjoint nonempty open subsets.

What would happen if we would consider one-point set as neither connected nor disconnected (or both connected and disconnected)?

Is this situation similar to "should $1$ be considered prime or not"?

Answer 1

Your first paragraph conflates "connectedness" and "path connectedness"; they're not the same.

For the second paragraph, this is like asking, "What if we said 2 was the same as $\pi$?" Your proposed "thing to say" might be amusing, but it contradicts both of the definitions of connectedness, for all possible topologies. So it's not very helpful.

I guess it's a little like the "is 1 prime" question, in the sense that there appears to be a widely accepted definition with which you have some qualms.

Let me add a little detail. We could say "a set $X$ is connected if (a) $X$ does not consist of a single point, and (b) for every pair of disjoint open sets $U$ and $V$ such that $U \cup V = X$, one of $U$ or $V$ is empty." And we could define "disconnected" similarly, and then observe that one-point sets are neither connected nor disconnected.

The result would be that virtually every proof about connectedness would have an extra paragraph to say something about 1-point sets, or exclude 1-point sets in some way. I don't believe that this would lead to greater insight.

Answer 2

As you've correctly demonstrated, the definition of connectedness resp. path-connectednes applies perfectly fine to the one-point space. So there is really no need to make an exemption. I also would not draw the analogy to the Is $1$ prime?- Question because in this case there are just no arguments for the one-point-space not to be path-connected.

Moreover one likes to think about path-connectedness as a property not only of spaces but also of homotopy classes of spaces. (https://en.wikipedia.org/wiki/Homotopy#Homotopy_equivalence) For example the one-point-set is homotopy equivalent to $\mathbb{R}^n$ (and by definition also to every other contractible space). So if you want to consider euclidean space as path-connected, then you should also consider the one-point-space as path-connected.