We know that in Hausdorff space singleton's are closed and also that limits are unique. Does it mean that for a subspace A, the set of limit points of the subspace contains the unique limit? In other words, the set of the limit points of A is a singleton, which is the only point that every sequence in the space converges to.
I believe this is wrong, and I might have misinterpreted the idea of limit points.
Any help would be appreciated.
Thank you!
Best Answer
No, take e.g. $A=(0,1)$ in the reals (usual topology). Here $A'=[0,1]$, i.e. all points of $[0,1]$ are limit points of $A$: every neighbourhood of a point $x \in [0,1]$ contains a point $y \in (0,1)$ with $y \neq x$ (this is the definition of a limit point). In a metric space this implies that there is a sequence in $(0,1)$ (of all distinct points, even) that converges to this $x$, and this fact is probably the origin of the term "limit point": the point can be a limit for a seqeuence in $A$, but of course there are many sequences in $A$ and many possible limits of such sequences. In a Hausdorff space we have that if a given sequence converges there can be only one point it converges to and you remember this fact as "limit points are unique". But this does not mean a limit point of a set is unique, only a possible limit of a sequence is; these are different notions that you should separate in your mind. As I said, in a metric space there is a relation in the sense that a limit point of a set is a limit of some sequence from that set, in general spaces that doesn't need to hold, though. My example shows that the set of limit points can be large (even much larger than the set we start with, e.g. the set of limit points of $\Bbb Q$ is the whole of $\Bbb R$) and is only rarely just a singleton.