Let $X$ be a topological space, $(x_\alpha)$ a net in $X$ and $A \subseteq X$ an arbitrary subset.
The point $x \in X$ is
- a cluster point of $x_\alpha$ if for every neighborhood $U$ of $x$ the net $x_\alpha$ is frequently in $U$ or equivalently if $x_\alpha$ has a subnet that converges to $x$.
- an accumulation point of $A$ if every neighborhood $U$ of $x$ contains a point of $A$ distinct from $x$, i.e. $|U \cap A \setminus \{ x \}| \geq 1$
- an $\omega$-accumulation point of $A$ if every neighborhood $U$ of $x$ intersects $A$ in an infinite amount of points, i.e. $|U \cap A| \geq \aleph_0$
- a complete accumulation point of $A$ if for every neighborhood $U$ of $x$ it holds $|U \cap A| = |A|$, i.e. every neighborhood intersects $A$ in as many points as $A$ has (measured by cardinality).
(Some incidental remark: if $X$ is $T_1$ then any accumulation point is an $\omega$-accumulation point.)
There are some relations between cluster points of sequences and $\omega$-accumulation points of the range of the sequence: Every $\omega$-accumulation point of $\{ x_n \}$ is a cluster point of $(x_n)$ and vice versa, if all the $x_n$ are eventually distinct then any cluster point of $(x_n)$ is an $\omega$-accumulation point of $\{ x_n \}$.
Also, if $A$ is countable then any bijection $x : \mathbb{N} \to A$ gives us a sequence $(x_n)$ with distinct points such that $A = \{ x_n \}$ and we can relate cluster points of $(x_n)$ with $\omega$-accumulation points of $A$.
Questions:
- Are there some relations between nets $(x_\alpha)$ and (complete) accumulation points (or other types of accumulation points) of the ranges $\{ x_\alpha \}$?
- If $A$ is an arbitrary (not necessarily countable) set, can we then write $A = \{ x_\alpha \}$ for some net in $A$ and relate cluster points of $x_\alpha$ with (complete) accumulation points of $A$?