Some problem about first countability (general topology) I have read some books in topology, they usually discard the finite case of first countability axiom (in those books, countable=finite+countably infinite), and then went on to claim that we can construct an indefinitely shrinking neigbourhoods by taking intersection. However, this does not hold if there is only finitely-many members in a local base. Is it true that most of the authors just try to avoid this kind of 'trivialities'?
 A: Without being able to read the text you are using, it is hard to say if this particular author is avoiding trivialities. It is certainly true that some authors do avoid trivialities, but some will explicitly note and deal with the trivialities. I have two guesses on why you are seeing this in your text.
$(1)$ The word "countable" can vary in meaning from author to author as well. The definition of countable I was first given was applicable to finite and countably infinite. Yet when I was working through theorems in my text, any time the word "countable" appeared it was strictly meant to be countably infinite. Otherwise the theorem would explicitly include the word "finite". Is it possible the author of your book is doing the same? If so, the claim you mention in your post would make sense.
$(2)$ The author might be using a softer definition of "shrinking". Letting $X$ be a space containing $x$ such that $x$ has a finite local basis $\mathcal{B} = \{B_i\}_{i=1}^n$, then we know one of the $B_i$'s is a subset of every open set containing $x$. Let's say $B_n$ is this element. Then $$B_1 \supset B_1 \cap B_2 \supset \ldots \supset \bigcap_{i=1}^nB_i \supset \left(\bigcap_{i=1}^nB_i\right) \cap B_n \supset \left(\bigcap_{i=1}^nB_i\right) \cap B_n \cap B_n \supset \ldots $$ can be seen as an indefinitely "shrinking" sequence of neighborhoods even though eventually you can replace each $"\supset"$ symbol with $"="$ (I think of it as being analogous to a constant sequence that is trivially monotone decreasing). Anyway, that means the sequence $\{B_1,B_2,\ldots,B_{n-1}, B_n,B_n,B_n,\ldots\}$ is countably infinite and satisfies the first countability axiom. This is all pretty trivial, and after writing all of this up I could see why an author wouldn't want to!
