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It appears that some topologists (in particular those who I am learning from) really like nhoods but not so much open sets. I say this because their definitions always involve nhoods where they could equivalently (and seemingly more intuitively) involve just open sets.

Here I take a nhood of $x\in (X,\tau)$ to be any subset of $X$ containing an open set that contains $x$.

For example:

Definition: $X,Y$ are two topological spaces. A function $f:X\rightarrow Y$ is continous at $x\in X$ iff for every nhood $V$ of $f(x)$ there is a nhood $U$ of $x$ such that $f(U)\subseteq V$.

But an equivalent definition arises when we replace 'nhood of' with 'open set containing' (i.e. instead of nhood of $x$ we say open set containing $x$). This seems much more natural to me, which leads me to ask: am I missing something here? Why/is it better to use nhoods in definitions instead of open sets.

To me nhoods seems rather clumsy compared to open sets.

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fosho
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  • They are just equivalent. As it is, you can choose the most "useful" one.. – Paolo Leonetti Mar 23 '17 at 20:21
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    Sometimes a "neighborhood" is assumed to be an open set in the first place - it depends on the author. – Math1000 Mar 23 '17 at 20:43
  • Do you happen to know if it is more common to prefer open sets to nhoods in the literature? – fosho Mar 23 '17 at 20:49
  • thats interesting question! I have had essentially the opposite question: https://math.stackexchange.com/questions/2846861/why-is-continuity-characterized-with-open-sets – Charlie Parker Jul 10 '18 at 17:31

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It is a direct generalization of the $\epsilon - \delta$ definition of continuity on metric spaces, so it might be considered easier to learn. For example the definition of continuity of a function from $\mathbb{R}$ to $\mathbb{R}$ in terms of open sets is usually motivated by the fact that it is equivalent to the more intuitive epsilon-delta definition.

mathematician
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  • wouldn't basic nhood be the generalization here? Or even basic open sets... – fosho Mar 23 '17 at 20:27
  • Yes, that's what I had in mind. For a more in depth version of my answer see the second answer to: http://math.stackexchange.com/questions/157735/definition-of-neighborhood-and-open-set-in-topology?rq=1 – mathematician Mar 23 '17 at 20:42
  • Ah, thanks for the link :) – fosho Mar 23 '17 at 20:47