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Let me start with introducing concepts of what acc pt and lim pt are.

Adherent point of X is the point whose every neighborhood contains some points of X.

And limits point of X is the point whose every neighborhood contains some points of X distinct from the given point.

I already know that X is closed if and only if it contains all its adherent points or

It contains all its limit points.

At the definition of limit, we usually use the concept of limit point.

Then why we need concept of adherent point?

It seems to me that it suffices to define limit or some other concept with limit point.

It looks redundant to have adherent point.

Although I know that they are very different concept from each other, I cannot see the reason why we need adherent point.

Could you help me to explain why we need such concept?

Thank you for your kind reply in advance.

glimpser
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  • For example, the notion of "perfect set" requires the notion of limit point (in the terminology you use here). – Ian Feb 11 '18 at 04:51
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    I've never seen the characterization of an accumulation point that you give. I've only ever seen the two terms (limit point and accumulation point) used interchangeably. Where are you seeing this distinction being made? – Xander Henderson Feb 11 '18 at 05:01
  • This definition of accumulation point does not seem to be very interesting. It is equivalent to be a limit point of $X$ or an element of $X$ (inclusive 'or'). As hinted by other comments your definition does not seem to be widely used. – Clément Guérin Feb 11 '18 at 05:05
  • That being said, this nearly identical question has several answers, many of which make (some rather fine) distinctions between different kinds of limit points. – Xander Henderson Feb 11 '18 at 05:05
  • @XanderHenderson sorry I mis-typed accumulation point as indicating adherent point. – glimpser Feb 11 '18 at 05:38
  • @Grimza Well, that makes a world of difference. Though that question is also addressed in the answers to the question that I linked above. – Xander Henderson Feb 11 '18 at 05:41
  • @XanderHenderson I'd like to know why we need adherent point. – glimpser Feb 11 '18 at 05:42
  • @Grimza If you want an answer to that question, a good first step would be editing your question to use the correct terminology. – Xander Henderson Feb 11 '18 at 05:57
  • I also feel the same about the term "adherent point" and +1 there. From the few textbooks I have read the terms "accumulation point" and "limit point" are same and the first one conveys the meaning of the term more accurately and the second one is easier to type. – Paramanand Singh Feb 11 '18 at 08:48
  • Adherent points are introduced because the closure of $A$ is the set of adherence points of $A$. – Henno Brandsma Feb 11 '18 at 22:00
  • @HennoBrandsma I know that definition, but isn't it more efficient that define the closure of A by limit point, and so the others are? – glimpser Feb 11 '18 at 23:33

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