The identity theorem of complex analysis relates to the extrapolation of the equality of two complex functions between domains. In the statement, it is required that one domain have atleast one accumulation point in the other for this statement to work.
I am aware of density and it's multipile uses in analysis. A subset $S$ of topological space $X$ is said to be dense in the aforementioned space, if for every open set about any point in $x$, there is a non empty intersection with the subset $S$. This is quite a strong fact, as it means all points of $X$ can be approximated through $S$.
An example of this idea being realized in Analysis can be seen in the following:
Stone Weierstraß theorem: Every continous function can be approximated by a polynomial
Measure Theory (Indirectly): Every continous function can be approximated by a sum of simple functions Refer
Transfering countability of basis in a metric space: If we have that $S$ is a countable set, then we can prove that we can find a countable basis for $X$ through application of density.
These facts stem from the idea that we can approximate any point in X in a topological sense through points of $S$.
I think maybe a way to think about only a single accumulation point is to consider it as flipping the script. This one accumulation point can act as a stand in for the entire set $S$ when discussing it's properties in $X$.
I'm trying to understand, what is the topological significance(/implication) of having a single accumulation point? Or perhaps, explain how it is useful in putting various proofs?