When looking at Wikipedia article about semi-continuity I saw equivalent condition for upper-semicontinuity using limit superior. However, I think that the condition that $X$ is metric is redundant there.
The following is an exact quote from the current revision of the Wikipedia article.
We say that $f$ is upper semi-continuous at $x_0$ if for every $\epsilon > 0$ there exists a neighborhood $U$ of $x_0$ such that $f(x) \leq f(x_0) + \epsilon$ for all $x \in U$ when $f(x_0) > -\infty$, and $f(x)$ tends to $-\infty$ as $x$ tends towards $x_0$ when $f(x_0) = -\infty$.
For the particular case of a metric space, this can be expressed as $$\limsup_{x\to x_{0}} f(x)\le f(x_0)$$ where lim sup is the limit superior (of the function $f$ at point $x_0$). (For non-metric spaces, an equivalent definition using nets may be stated.)
There is also a corresponding statement about lower semicontinuity and limit inferior.
When I try to prove this characterization or look at proofs elsewhere (for example, Upper semicontinuous functions) I do not see that the assumption that $X$ is metric is needed anywhere. Moreover, sequences or nets do not appear in the statement at all, which makes me even more suspicious.
I understand that metrizability (or some countability axiom) would be needed to get the characterization of upper semicontinuity at the point $p$ in the form $$f(p) \ge \limsup\limits_{n\to\infty} f(x_n)$$ for every sequence $x_n\to p$. (And in this formulation it would make sense to replace sequences by nets.) For example, in this post the characterization using sequences is mentioned for first countable spaces: Characterization of lower semicontinuous functions
Am I right that the requirement for $X$ to be metric is not needed there? (I guess that it is possible that the Wikipedia article might be edited as a result of this question. I suppose that Wikipedia editor might be thankful for a reference to a book containing these claims. And so would I.)