Is this an appropriate use of the word 'trivially'?

Question

My adviser is taking issue with my use of the word trivially, seen below. I cut out some nonessential stuff.

A set $E$ is closed if every limit point of $E$ is a point of $E$.

A set $E$ is perfect if $E$ is closed and if every point of $E$ is a limit point of $E$.

Consider the set $\{0,1\}$, which is closed (trivially) because it has no limit points. This same fact implies that $\{0,1\}$ is not perfect.

My adviser says that I should never use the word 'trivial'. I agree that this is a good rule of thumb, and generally I am more likely to misuse it than someone with more experience. However, I think the situation above is one case where the use of the word 'trivial' is justified.

The set $\{0,1\}$ is closed, yes, but why? Not because it contains its limit points, but because it doesn't even have limit points. I think that it actually adds information to tell the reader that this is trivial, whereas in other cases saying something is 'trivial' is either lazy or not needed.

Answer 1

I'd say that using the word "trivially" is a context-dependent kind of thing. In particular: let's say you were in the middle of a long proof, and one step amounts simply to applying a definition. That one step might be considered trivial.

In this case, the entire question comes down to grappling with definitions, so calling it "trivial" doesn't feel right. I like Ovi's suggestion of "vacuously", since we're addressing a sort of degenerate case. Here's what I would write:

Consider the set $\{0,1\}$. It is closed because it is finite and thus has no limit points. But since it closed with no limit points, $\{0,1\}$ is not perfect.

Answer 2

I don't like "vacuously." A statement is vacuously true if it is of the form $P\Rightarrow Q$ with $P$ is false. This is not of that form.

I would take out the word "trivially" and leave the rest of the sentence as-is.