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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:

  1. Are there some relations between nets $(x_\alpha)$ and (complete) accumulation points (or other types of accumulation points) of the ranges $\{ x_\alpha \}$?
  2. 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 $A$?
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  • $\begingroup$ Problem is, that for uncountable image sets there are very many index sets and nets that will have the same range. The range is way too coarse compared to the net structure (which includes the order on the index set!). What kind of result are you aiming for? $\endgroup$ – Henno Brandsma Feb 4 '16 at 21:54
  • $\begingroup$ @HennoBrandsma I was hoping that such a question seems to be a natural extension of relations in the countable case and thus that there should be already some knowledge available. What makes the big difference between the countable case and the uncountable one? Are there some relations when restricting the considerations to linearly ordered nets (ordered by an ordinal) or cofinal subnets? $\endgroup$ – yadaddy Feb 5 '16 at 9:21
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    $\begingroup$ @HennoBrandsma In particular, I met in the literature several definitions for "relative compactness" of a subset $A \subseteq X$ some of which are equivalent for general topological spaces and some of which require additional separation properties to be equivalent (e.g. "closure is compact" or "contained in a compact subset" (Bourbaki) or "every net has a cluster point in the ambient space" (Archangelskii)). Is the following statement true: "every infinite subset of $A$ has a complete accumulation point in $X$" if and only if "every net in $A$ has a cluster point in $X$? $\endgroup$ – yadaddy Feb 5 '16 at 9:27

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