I'm currently taking a course on analysis over $\mathbb{R}^n$ and the book used defines that a point $p \in \mathbb{R}^n$ is a limit point of a set $X\subset \mathbb{R}^n$ if $p$ is the limit of some sequence of points of $X$. Now, because of this the author says "well, any point of $X$ is the limit of the constant sequence with every element equal to the point, so that any point of $X$ is a limit point of $X$". After that the author defines the closure of a set $X$ as the set $\overline{X}$ of all limit points of $X$ and the author states that $X\subset \overline{X}$.

Now, in the same time I am studying metric spaces on Rudin's Principle of Mathematical Analysis. In this book, the author defines that in a metric space $M$, a point $p \in M$ is a limit point of a subset $X\subset M$ if every open ball with center at $p$ contains some point $q \in X$ with $p \neq q$. Then with this definition of limit point the author defines closure again as the set of all limit points.

In truth the first author gives the name "adherent" points for limit points, but it seems that they are talking about the same things (since the only difference is the definition of limit point, the definition of closure, closed set and so on are the same).

The only problem is that with the second definition, $X\nsubseteq \overline{X}$ because $X$ can have isolated points. It suffices to consider $S^1 \cup \{(0,0)\}$. The point $(0,0)$ is in the closure according to the first definition, but not in the closure according to the second one (every open ball with radius $< 1$ misses points of $S^1$).

Are these notions different, is there some mistake made by some of the authors, or I am not really understanding that we are working with different situations?

Thanks a lot in advance for the help!

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    $\begingroup$ Adherent points are not to be confused with limit points. And I'm pretty sure that Rudin doesn't define the closure of $X$ as the set of limit points, rather as the union of $X$ with the set of $X$'s limit points. $\endgroup$ – Daniel Fischer Aug 14 '13 at 21:32
  • $\begingroup$ Sorry, I forgot this point. That's true, he defines as the union. Now is that the difference? One adherent point is the same as a point that is either in the set or in the set of limit points? Thanks for the help! $\endgroup$ – user1620696 Aug 14 '13 at 21:34
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    $\begingroup$ It would violate the Kuratowski Closure Axioms to have $X\not\subseteq \overline X$, by the way. $\endgroup$ – Karl Kronenfeld Aug 14 '13 at 21:35
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    $\begingroup$ Yes. An adherent point of $X$ is a point $x$ such that every neighbourhood of $x$ intersects $X$. If $x \in X$, that is automatically the case. So $x$ is an adherent point of $X$ if $x \in X$ or $x$ is a limit point of $X$ (note: the two aren't mutually exclusive). $\endgroup$ – Daniel Fischer Aug 14 '13 at 21:36

Seconding @DanielFischer's comments, but perhaps making a larger point:

First, one cannot absolutely rely upon consistent terminology across sources, across time, especially (!) from textbooks, where an exaggerated precision is often the style. Thus, we have to make inferences from context.

Second, for many decades, I think there have been no exceptions to $X\subset \overline{X}$, regardless of use of words about "limit point" and "adherent point" (the latter even less standard, in my experience). So, in fact, in your two different contexts, the words' meanings have to be interpolated to make this desired fact correct.

Third, indeed, in a world where there are typoze, the best way to be sure about edge-cases of "definitions" is to look at the intention, if it is admitted anywhere near the literal definition, e.g., in examples, best.

An example of a non-issue is "whether or not 1 is a prime". The genuine issue is that it doesn't matter much what we make the words mean, but correct statements become messier if "1 is a prime, by definition", rather than "1 is not a prime, by definition". Thus, after a few centuries of varying opinions/conventions about this, currently "1 is by definition not a prime". But, the point is, the underlying reality did not change because a semantic convention changed. The two deserve separation.


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