If the set of integers were given the indiscrete topology wo

If the set of integers were given the indiscrete topology, would it be connected? Compact? Hausdorff?  

The answer is It is connected, compact but not Hausdorff. But can you explain why?

Solution

in indiscrete topology the only open sets are S, phi.

and also these are the only closed sets.

so by theorem we know that X is connected if The only subsets of X which are both open and closed (clopen sets) are X and the empty set.

hence it is connected.

X is a hausdroff if any two distinct points of X can be separated by neighborhoods.

since there are only two set one is X and another is empty set,

so we can not seperate two distinct points of X by their neighbourhood.