Is a singleton in a p-adic space connected?

Question

I want to use a theorem with a p-adic space:

The image by a continuous epimorphism of a connected space, is itself connected.

Correct me if I'm wrong, but I think the fact that every p-adic field is totally disconnected, is a fairly major stumbling block, rendering the theorem nullipotent in p-adic spaces.

BUT... I'm confused by this:

Am I right in thinking that a singleton is connected?

The rule I have is that a space is connected if it cannot be represented as the union of two or more disjoint non-empty open subsets.

Clearly a singleton cannot be represented as the union of two or more disjoint non-empty subsets, irrespective of whether or not they're open, so must by this rule be connected.

Then we can conclude there can be no continuous, noninjective epimorphisms in a padic space which project multiple points down to any singleton.

Answer 1

Singletons are connected in every space. A space is called totally disconnected if the singletons are the only connected subspaces. What follows is that a continuous map from a connected space $X$ to a totally disconnected space $Y$ is constant; in particular, it's non-injective unless the domain $X$ was a singleton to begin with.