I have a major problem in understanding connected set!
Consider the set $X = (0,1]$. Now lets describe the following sets which can be considered as the basis for the topology on set $X$.
1) For all $b\in X$, all sets of the form $(0, b]$
2) For all $b\in X$, all sets of the form $(b, 1]$
Clearly above 2 groups of sets together can form the basis of the topology on set $X$. Now clearly we can $(0, x]\cup(x,1]$ is the disjoint union of the set $X$.
In the textbook it is written $(0,1]$ as connected BUT we can see it can be expressed as disjoint unions of open sets as above, Hence not connected! Kindly help.
PS: You can argue that sets of $(0, x]$ nature are not open, but they form the topology on $X$, hence they are open.