In topology there is the notion of an infinite spider's web in the (complex) plane $\mathbb C$ that was introduced in 2010 https://arxiv.org/pdf/1009.5081.pdf.
A set $E\subseteq \mathbb C$ is an infinite spider’s web if $E$ is connected and there exists a sequence of bounded simply connected domains $(G_n)$ with
- $G_n \subset G_{n+1}$,
- $\partial G_n\subset E$, and
- $\bigcup _{n\in \mathbb N}G_n = \mathbb C$.
In certain cases we also have that $E$ is closed and nowhere dense, and each $\partial G_n$ is a simple closed curve (Jordan curve), so that $E$ more closely resembles a traditional spider's web. These sets can be generated by iteration of entire functions such as $f(z)=\frac{1}{2}(\cos z^{1/4}+\cosh z^{1/4})$. The image below shows a spider's web consisting of the points $z\in \mathbb C$ such that $f^n(z)\to\infty$ at a certain rate (see https://arxiv.org/pdf/1009.5081.pdf for details).
Every locally connected Julia set of a transcendental entire function also has this form (shown in https://arxiv.org/pdf/1110.3256.pdf).
