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How can I define a neighborhood (not only $\delta$ neighborhood) of a point in $\mathbb R$.

without using metric concept.

According to rudin's definition which must be a open set.

Supriyo Banerjee
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  • If $\mathcal T $ is a topology on $\Bbb R $, a neighborhood of a point $a$ is any open set $U$ (that is $U\in \mathcal T $) containing $a$. – cqfd Jan 12 '19 at 11:54

1 Answers1

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You need some concept. If you don't want metric concept, then you at the very least need topology. And in topology, rather than a metric, all you have available to work with is the collection of all subsets of $\Bbb R$ which we call "open".

In this context, a neighborhood of $x\in\Bbb R$ is simply any such open set which contains $x$.

Arthur
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    A small clarification: in some books any set which contains an open set containing $x$ is called a neighborhood of $x$. – Kavi Rama Murthy Jan 12 '19 at 12:06
  • @KaviRamaMurthy exactly.In many undergrad books it is written. – Supriyo Banerjee Jan 12 '19 at 12:11
  • @KaviRamaMurthy You're right. With that convention, an open neighborhood is an open set containing the point, and a neighborhood is any set which contains an open neighborhood. – Arthur Jan 12 '19 at 14:05