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Always, when working on Topology and taking a neighborhood of a point $x$, we end up using an open set $G$ that contains $x$. Why, then, mathematicians define the concept of neighborhood if always we end up working with open sets? How can we exploit that generality?

HeMan
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    Not always. I have seen the term "compact neighbourhood" used, for instance (though I can't remember where...). It's just that so much of the machinery of topology relates to open sets, which makes them easy to focus on. – Arthur May 25 '17 at 04:20
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    A neighborhood of $x$ is by definition an open set containing $x$. What do you think of when you hear open set versus neighborhood? – Alex Ortiz May 25 '17 at 04:21
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    @AOrtiz No, the most common definition of neighbourhood does not require the set to be open, just to contain an open set containing $x$. See e.g. Wikipedia – Robert Israel May 25 '17 at 04:31
  • I ' ve noticed the same and think for the most part it's a useless historical hangover from when topology was being developed. I suggest you have a look at neighborhood spaces. Of additional interest may be networks. – William Elliot May 25 '17 at 04:45
  • @WilliamElliot It allows for formulations like: a space is locally compact iff every point has a compact neighbourhood (or a local base of compact neighbourhoods, if you fancy a stronger version, equivalent in Hausdorff spaces). Or space is $T_3$ iff every point has a local base of closed neighbourhoods. – Henno Brandsma May 25 '17 at 07:25

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