Closed sets and limit point confusion The definition of limit point of a set is that for x to be a limit point of a set, every neighbourhood  of x must contain a point different from x itself.
A closed set is defined to be a set which contains all its limit points.
By this logic it seems that the open interval (0,1) should be closed since every point has all their neighbourhoods containing an element different from itself
My question is where am i wrong??
 A: The key here is that there is a limit point for $(0,1)$ which does not lie in $(0,1)$. You've stumbled on the fact that every set is closed in itself, but might not be closed in some larger set.
In this case, the limit points which $(0,1)$ does not contain are $0$ and $1$. For instance, every open neighborhood of $0$ contains some small positive number, which will lie in $(0,1)$.
A: 
since every point has all their neighbourhoods containing an element different from itself

That means that every point in $(0,1)$ is a limit point.
But that doesn't mean every limit point is in $(0,1)$.
......
$0$ and $1$ are limit points of $(0,1)$ because every neighborhood of $0$ will contain a point larger than $0$ that is in $(0,1)$ and every neighborhood of $1$ will contain a point less than $1$ that is in $(0,1)$.
So $(0,1)$ does not contain two of its limit points.  So it is not closed.
.....
Bear in mind, a limit points definition says every neighborhood of the point as a point other than itself that is in the set.  It does NOT say that the point itself is in the set (although if it isn't, it must logically "be on the edge" of the set as every neighborhood of the point will have point(s) in the set.
If any limit point is outside the set then the set is not open.
Also not, points in a set (closed or not) needn't be limit points.  The set $[0,1]\cup \{3,4\} \cup [4.1, \infty)$ is closed but $3,4$ are not limit points.
A: "The definition of limit point of a set is that for $x$ to be a limit point of a set, every neighbourhood of $x$ must contain a point different from $x$ itself."
This is incorrect. The definition of "limit point" is that every neighborhood of $x$ contains a point of $S$ other than $x$. So if your frame of reference is $\Bbb{R}$ in the usual topology, $S = (0, 1)$ does not contain $0$ and $1$, which are limit points of $S$. So $S$ is not a closed set.
But if your frame of reference is $(0, 1)$ in the subspace topology from $\Bbb{R}$, then yes, $(0, 1)$ is a closed set with regards to that topology.
So the errors are:

*

*stating the definition of a limit point

*assuming that a set which is open/closed with respect to a certain topology, is open/closed w/r/t any topology that can be placed on it

