Let $X$ be a topological space. $Y$ is a discrete subset of $X$ if it has a discrete topology induced by the topology of $X$. This is equivalent to the fact that for every $y\in Y$ there is an open $U\subset X$ such that $U\cap Y=\{y\}$. Consider a stronger condition: there is a collection $\{U\_y\}\_{y\in Y}$ of mutualy disjoint open sets, such that $y\in U\_y$. I wonder if the two are equivalent for nice spaces, but my question today is following:



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> Is it true that if $X$ is locally compact and not compact, there is a discrete (in the stronger sense) set $Y\subset X$, such that $\overline{Y}$ is not compact?
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In my specific case $X$ is locally compact, but this condition is not necessary in order to pose the question. If $X$ is metrizable and separable, it has a metrizable compactification, and $Y$ can be chosen as a sequence that converges in a conrolled way to an infinite element. If $X$ is locally compact and $\sigma$-compact, then there is a sequence of compacts $\{K\_n\}$ such that $K\_n\subset int~K\_{n+1}$, and taking $Y$ to be the sequence of arbitrary $y\_n\in int~K\_{n+1}\backslash K\_{n}$ will work.


I also have a reference request about existence of an infinite discrete subset in any infinite Hausdorff space. This follows from the result in [this article,](https://arxiv.org/abs/1712.08739) but I was not able to find a more direct reference. I found some research on this topic, but not this explicit statement.



Existence of a discrete subset
The answer to this question is negative and can be obtained with the help of weak P-points in compact spaces with countable cellularity. A non-isolated point $p$ of a topological space $X$ is a *weak $P$-point* if $p$ is not an accumulation point of a countable set $C\subset X\setminus\{p\}$.


If $p$ is a weak P-point in a compact space $X$ with countable cellularity, then $X\setminus\{p\}$ is a locally compact space such that any strongly discrete subset is countable and hence has compact closure in $X\setminus\{p\}$.


It seems that examples of countably cellular spaces with weak P-points can be found using [this result of Dow.](http://www.ams.org/journals/tran/1982-269-02/S0002-9947-1982-0637709-4/S0002-9947-1982-0637709-4.pdf) 


