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From what I can gather, a limit point is a point $a \in A$ such that the open ball $B(a; \delta)$ contains a point other than $a$. While an isolated point is a point $b \in A$ such that the open ball $B(b;\delta)$ contains no point different from $b$. So, is every point always a limit point or an isolated point?

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Once you clarify the quantifiers on $\delta$, your answer will be YES. –  GEdgar Nov 23 '12 at 0:13
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up vote 3 down vote accepted

You should be a lot more careful with your definitions, there are a couple important parts missing:

  1. Let $X$ be a metric space and $A\subset X$. A point $a\in X$ is said to be a limit point of $A$ if for all $\delta>0$, there is an $x\in A\cap B(a;\delta)\backslash\{a\}$.

Notice: a limit point need not be in $A$! Example: $A=(0,1)\subset\Bbb{R}$. Both $0$ and $1$ are limit points of $A$.

  1. Let $X$ be a metric space and $A\subset X$. A point $b\in A$ is an isolated point of $A$ if there exists $\delta>0$ such that $A\cap B(b;\delta)\backslash\{b\}=\emptyset$.

Notice: isolated points must be in $A$.

So to answer your question, it is easy to show that $b\in A$ is an isolated point of $A$ if and only if $b$ is not a limit point of $A$. So if you only consider points in $A$, then the answer to your question is yes. However the statement "a point is either a limit point or an isolated point" is technically false, since it is possible to be neither if your point is not in $A$ (e.g. take $x=2$ with $A=(0,1)$; $x$ is neither a limit point nor an isolated point of $A$).

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THank you so much! This clarifies alot! –  Eric Nov 23 '12 at 1:05
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In your definition, you forgot the important part: the $\exists$ and the $\forall$. Once you clarify this, as @GEdgar said, the answer will clearly be yes.

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A limit point of a set $B$ is a point $a$ such that EVERY open ball about $a$, no matter how small, contains some point in $B$ other than $a$ itself. A limit point of $B$ need not be a member of $B$, but may be. You can't omit the word "every".

An isolated point of $B$ is a point $a$ that is a member of $B$, but such that some open ball centered at $a$ contains no point of $B$ other than $a$.

A point $a$ that does not belong to $B$, and is the center of some open ball that does not intersect $B$, is neither a limit point of $B$ nor an isolated point of $B$.

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