Talk:Zeroes of Analytic Function are Isolated

So let $k$ be the least number such that $a_j = 0$ for $0 \le j < k$, and $a_k \ne 0$.

This assumes that not all $a_j$ are 0, right? Would it be reasonable to add a note explicitly stating that if all $a_j$ are 0, then $f$ must be a constant function (which is one of the results of the theorem)? --Aaron1011 (talk) 19:57, 4 November 2021 (UTC)

totally disconnected → discrete

I want to keep the page title and edit this part. See Topological Space is Discrete iff All Points are Isolated and Discrete Space is Totally Disconnected. --Fake Proof (talk contribs) 07:37, 21 July 2022 (UTC)

I'm not so sure. Despite the fact that the complex field under the usual metric is a topological space, I'm discomfortable about mixing mathematical fields unnecessarily. This is a result purely in complex analysis, where isolated points are well enough understood, but it's an unnecessary burden to expect students of complex analysis to take on board results from point set topology for no particular reason.

In this case, the page title refers to isolated points, and the result gives isolated points, but the statement of the theorem talks about a disconnected space. No reason for this except the whim of the contributor who was part of the initial strategy of "let's just post up random stuff we know" with no thought to site structure. He also had no interest in the concept of links to established results. We moved on from that shambolic mess of an approach, but some of those initial pages were never properly revisited and rationalised.

If we do indeed specifically need the topological result about disconnected spaces, it makes more sense to add a new page for this, whose proof (while being essentially trivial) will use use topological terminology and results.

Mind, having said that, the concept of an "isolated zero" is already a topological concept which also needs definition in the concept of the complex plane, but as a concept it is more directly intellectually accessible than the concept of the zeroes forming a subspace which is "totally disconnected".

Keep it simple. --prime mover (talk) 08:41, 21 July 2022 (UTC)