Let $X$ be a compact Hausdorff space and $p\in X$ be a non-isolated point. Is it always possible to find a net $(x\_\alpha)\_{\alpha\in (I,\leq)}$ in $X\setminus\{p\}$ converging to $p$ such that $(I,\leq)$ is *totally ordered* (or a regular cardinal, which gives you the same thing)?


I would expect this to be false, but I do not know how to construct a counter-example.
I tried some standard spaces like the $1$-point compactification of an uncountable discrete space (because here the infinite point has no countable basis of neighborhoods) or a product of continuum many copies of $\{0,1\}$, but none of the worked. Probably the Stone-Cech-Compactification of the integers could be a counter-example but I really have no exprerience in working with that space ... Any ideas?



When can we find a net, defined on a totally ordered index set, converging to a non-isolated point in a compact Hausdorff space?
Yes. For every non-isolated $p\in X$, there is a well-ordered net in $X\setminus\{p\}$ converging to $p$ as long as $X$ is compact and Hausdorff. This result was observed in the 1992 paper Convergent free sequences in compact spaces by I. Juhász and Z. Szentmiklóssy, so let me paraphrase their argument. Suppose that $\lambda$ is a limit ordinal and $(A\_\alpha)\_{\alpha<\lambda}$ is a descending sequence of closed subsets of a compact Hausdorff space $X$ with $\bigcap\_{\alpha<\lambda}A\_\alpha=\{p\}$ but where $|A\_\alpha|>1$ for $\alpha<\lambda$. Let $x\_\alpha\in A\_\alpha\setminus\{p\}$ for $\alpha<\lambda$. Then whenever $U$ is a neighborhood of $p$, by compactness, there is some $\alpha<\lambda$ where $x\_\beta\in A\_\beta\subseteq U$ whenever $\lambda>\beta\geq\alpha$. Therefore, $x\_\alpha\rightarrow p$.


We still need to construct the sequence $(A\_\alpha)\_{\alpha<\lambda}$. Let $\lambda$ be the smallest cardinal such that there exists a collection of closed sets $(C\_\alpha)\_{\alpha<\lambda}$ where $\{p\}=\bigcap\_{\alpha<\lambda}C\_\alpha$ but where $p\in (C\_\alpha)^\circ$ for each $\alpha<\lambda$. We can therefore set $A\_\alpha=\bigcap\_{\beta<\alpha}C\_\beta$.


We also observe that some form of compactness is necessary for this argument to work; if $p\in\beta\mathbb{N}\setminus\mathbb{N}$, then there is no well-ordered net in $\mathbb{N}$ that converges to $p$.


**Unique accumulation makes the proof work**


Compact Hausdorff spaces can be characterized as the Hausdorff spaces where each net that accumulates at a unique point $p$ actually converges to the point $p$. This characterization of compact Hausdorff spaces allows us to break our result up into a couple of lemmas.


We say that a filter $\mathcal{F}$ on a topological space $X$ accumulates at the point $p\in X$ if $p\in\bigcap\_{R\in\mathcal{F}}\overline{R}$. We say that a net $(x\_{d})\_{d\in D}$ on the same space $X$ accumulates at the point $p\in X$ the filter generated by $\{\{x\_{e}\mid e\geq d\}\mid d\in D\}$ accumulates at the point $p$. Equivalently, the net $(x\_d)\_{d\in D}$ accumulates at $p$ precisely when $U\cap\{x\_e\mid e\geq d\}\neq\emptyset$ whenever $U$ is a neighborhood of $p$.


Lemma: Let $X$ be a Hausdorff space. Then the following are equivalent:


1. $X$ is compact.
2. If $(x\_d)\_{d\in D}$ is a net that accumulates at the unique point $p\in X$, then $(x\_d)\_{d\in D}$ converges to the point $p.$


Lemma: Let $X$ be a topological space. Let $p\in X$ be a non-isolated point. Suppose that $\{p\}=\bigcap\_{U\in\mathcal{N}(p)}\overline{U}$ where $\mathcal{N}(p)$ is the filter of neighborhoods of $p$ (this is a weakening of regularity). Then there is a cardinal $\lambda$ and well-ordered net $(x\_\alpha)\_{\alpha<\lambda}$ where if $(x\_\alpha)\_{\alpha<\lambda}$ accumulates at a point $q$, then $q=p$.


Proof: Let $\lambda$ be the smallest cardinal such that there exists a transfinite sequence $(U\_\alpha)\_{\alpha<\lambda}$ of neighborhoods of $p$ with $\bigcap\_{\alpha<\lambda}\overline{U\_\alpha}=\{p\}$. Then since $\lambda$ is minimized, the set $\bigcap\_{\beta\leq\alpha}\overline{U\_\beta}\setminus\{p\}$ is non-empty for each $\alpha<\lambda$. Then set $x\_\alpha\in\bigcap\_{\beta\leq\alpha}\overline{U\_\beta}\setminus\{p\}$ for $\alpha<\lambda$. Therefore, $\bigcap\_{\beta<\lambda}\overline{\{x\_\alpha\mid \alpha\geq\beta\}}\subseteq\bigcap\_{\beta<\lambda}\overline{U\_\beta}=\{p\}.$ $\square$


By combining the two above lemmas, we can conclude that for each point $p$ in a compact Hausdorff space $X$, there is a well-ordered net in $X\setminus\{p\}$ converging to $p$.


