Let $X$ be a topological space, and $x \in X$ be a point. There are two prevalent conventions on how to define a neighborhood of $x$:
Alternative Definitions (Neighborhood):
1) A neighborhood of $x$ is any open subset $W \subset X$ such that $x \in W$.
(This convention is used in Munkres's book for example.)2) A neighborhood of $x$ is a subset $W \subset X$ such that there exists an open set $A$ such that $x \in A \subset W$.
(For example this is the definition in Bourbaki's or Willard's "General Topology")
Thus, every neighborhood in the sense of (1) is a neighborhood in the sense of (2), but not vice-versa.
One often needs to show that a neighborhood of a point $x$ in the sense of (2) is actually open, and often this is a non-trivial verification from the given context. An example that I can come up with now (and this example was a motivation for asking this question) is the following:
Let $G$ be a topological abelian group, and $H$ a subgroup of $G$ which is also a neighborhood of $0 \in G$ in the sense of 2). Then one can show that $H$ is in fact an open set.
[The trick is to observe that for a given $g \in G$, the map $\phi_{g} :G \rightarrow G$ given by for $x \in G, \phi_{g}(x) = g + x$ is a homeomorphism, and so any neighborhood of a point $g \in G$ is of the form $g + U$, where $U$ is a neighborhood of $0$. Thus, for an $h \in H$, $h + H$ is a neighborhood of $h$, and moreover, $h + H \subset H$ because $H$ is a subgroup.]
The above proof shows that sometimes proving that a neighborhood in the sense of (2) is open is not completely trivial, while a neighborhood in the sense of (1) is always open. At times like these, I wonder why the second definition of a neighborhood is used at all. But I have learned topology primarily from Munkres, and so I might be ignorant of the advantages of definition (2).
So, what do you think are some of the advantages of using (2) as a definition for a neighborhood of a point $x \in X$, where $X$ is a topological space?
(This might be a duplicate question. But I searched a little bit and could not find a question that was exactly similar to this one. So, excuse me if I have asked something that was already asked.)