Notation for set of (open) neighbourhoods of a point. What is the most standard notation for the set of neighborhoods of a point $p$ in a topological space $X$?
I have seen $\mathcal{U}_X(p)$ and also $\mathcal{N}_X(p)$, but I never got the impression that those notations were standard. I couldn't find any answer to this in Google either.
And what about open neighborhoods?
Clarification: I am really looking for standard notations, so I hope that this question isn't interpreted as opinion-based.
 A: Note: There is no standard notation for the set of neighborhoods of a point $p$ in a toplogical space. A preference might be given by the mother tongue of mathematicians, if they tend to


*

*$\mathcal{U}_X(p)$ which indicates the first letter of the german term Umgebung or to

*$\mathcal{N}_X(p)$ which indicates the first letter of the english term neighborhood. 

Notational conventions from the great: A small selection of definitions from authors whose books are often regarded as classics.
  
  
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*Elements of Mathematics; General Topology, Chapters 1-4 by N. Bourbaki: We find in section 2: Neighbourhoods
Definition 4: Let $X$ be a topological space and $A$ any subset of $X$. A neighbourhood of $A$ is any subset of $X$ which contains an open set containing $A$. ... (and later:) Let us denote by $\mathcal{B}(x)$ the set of all neighbourhoods of $x$.
  
*General Topology by R. Engelking: We find in section 1.1: Topological spaces. Open and closed sets.  Bases. Closure and interior of a set.
If for some $x\in X$ and an open set $U\subset X$ we have $x\in U$, we say that $U$ is a neighborhood of $x$. 

We observe the term neighborhood by itself is not standard, since some authors define neighborhood which is defined by others as open neighborhood. This makes it even harder to consider a specific notation for the set of neighborhoods of $x$ as standard notation.

  
*
  
*General Topology by J.L. Kelley. We find in chapter 1: Topologies and neighborhoods in the paragraph before Theorem 1:
A set $U$ in a topological space $(X,\mathcal{T})$ is a neighborhood ($\mathcal{T}$-neighborhood) of a point $x$ iff $U$ contains an open set to which $x$ belongs. A neighborhood of a point need not be an open set, but every open set is a neighborhood of each of tis points. ... (and later:) The neighborhood system of a point is the family of all neighborhoods of the point.
Theorem 2: If $\mathcal{U}$ is the neighborhood system of a point, ...

Conclusion: First of all we should recall that the term neighborhood is not uniquely defined in the literature. We always have to check the definition used by the author. The different notation used by the authors above strongly indicate there is no standard notation.
A: As noted in the comments and the other answer, notation seems to vary widely. Instead, I will address the "other part" of your question-"...what about open neighborhoods?"
There tends to be a bit of waffling on this point-the reason being, one can define neighborhoods in terms of a topology, or define a topology based on a neighborhood system. This is much like defining equivalence relations in terms of a partition, or vice-versa.
Explicitly, we can define a neighborhood, if a topology has already been specified, to be a set containing an open set containing the point of which is it a neighborhood. As you can see, neighborhoods can be "very big", but the "larger" ones aren't of much interest to us, in general (topologies are "ranked", in terms of "niceness", in their ability to distinguish points at the "small" level-so, for example, the indiscrete toplogy is "very bad", i.e., not very useful).
On the other hand, if we define neighborhoods first, we can define a topology to consist of neighborhood sets which are neighborhoods of each of their elements. Either approach will lead us to the same place, eventually (convince yourself of this).
In analysis (rather than "pure topology") some preference is given for the neighborhood system approach, because the neighborhood system (at a point $p$) forms a filter, which lends itself well to generalizations of the concept of convergence in metric spaces. General topology tends to be (but not always) more concerned with "global" properties of the space, so the importance of particular points $p$ is not as great.
