# Thread: is this subset of R2 closed or open

1. ## is this subset of R2 closed or open

$\displaystyle X=\{(x,0) : 0\leq x\leq1\}$

i think it is closed, if we define a sequence : $\displaystyle x_{n}\in X, x_{n}=(\frac{1}{n},0)\rightarrow x=(0,0)\Rightarrow x\in X.$

i hope my definition of closed subset is right. Maybe my result is wrong coz i choose a special case.

another question ,what about $\displaystyle X=\{ (x,0) : 0<x<1\}$

2. Originally Posted by szpengchao
$\displaystyle X=\{(x,0) : 0\leq x\leq1\}$

i think it is closed, if we define a sequence : $\displaystyle x_{n}\in X, x_{n}=(\frac{1}{n},0)\rightarrow x=(0,0)\Rightarrow x\in X.$

i hope my definition of closed subset is right. Maybe my result is wrong coz i choose a special case.
It is closed because it contains all its limit points. It is not open as there are points in the set which are not in an open ball contained in the set.

another question ,what about $\displaystyle X=\{ (x,0) : 0<x<1\}$
Is neither open nor closed (in the usual topology anyway). It is not open as for any point in the set there is no open ball contained in the set which contains the point. It is not closed as it does not contain all its limit points.

CB