Why aren't all points in a set limit points (by this definition)? I am self-studying Real Analysis using S. Abbot's Understanding Analysis and am a bit confused with the definition of a limit point of a set on page 89. The definition is as follows:

A point $x$ is a limit point of a set $A$ if every $\epsilon$-neighbourhood $V_\epsilon(x)$ of $x$ intersects the set $A$ at some point other than $x$.

This confuses me a bit - doesn't this apply to all points in $A$? for example, if we defined some point arbitrary $a$ somewhere in the middle of the set, won't we be able to keep shrinking the size of the $\epsilon$-neighbourhood, and by archimedian property, always find a member of this $\epsilon$-neighbourhood that is a member of A but not $x$?
Can someone help me reconcile this? I am a bit confused. Also, the additional notion of isolated points make this even more confusing - perhaps someone could explain the differences as well?
Thanks!


This confuses me a bit - doesn't this apply to all points in A? for
example, if we defined some point arbitrary a somewhere in the middle
of the set, won't we be able to keep shrinking the size of the
ϵ-neighbourhood, and by archimedian property, always find a member of
this ϵ-neighbourhood that is a member of $A$ but not $x$?

No. Consider $\ A = [0,1]\cup \{ 2\}.\ $ For any $\ \varepsilon<1,\quad V_{\varepsilon}(2) = \{2\}.$
