Find the connected components of subspace of space with the particular point topology Let $A$ be a subspace of $X$ with the particular point topology. 
Find the connected components of $A$.
 A: Hint: Divide your analysis into cases depending on whether the subspace $A$ contains the particular point.  (In each case you should be able to give a simple description of the subspace topology.)
A: $S$, a subset of $A$, is open in $A$ iff it is empty or contains $x_0$.
$S$, a subset of $A$, is closed in $A$ iff it is $A$ or doesn't contain $x_0$.
So the only closed and open subsets are $\emptyset$ and $A$ because both $S$ and $S^{c}$ cannot contain $x_0$. Thus $A$ is connected.
EDIT : Oh, too fast. After seeing the upper comment, I realize that my answer is only valid if $x_0 \in A$.
RE-EDIT : Actually, the answer depends on how you understand the question :
X has the particular point topology, or is the topology induced by X on A the particular point topology ?
RE-RE-DIT : Ok, there's the continuation of the argument :
If $x_0 \in A$, then the induced topology on $A$ by $X$ is also the particular point topology, and then $A$ is, as I said, connected, so it has only one connected component, itself.
If $x_0 \not \in A$, then you have to show that the induced topology on $A$ by $X$ is the discrete topology, i.e. for every $B \subseteq A$, there is $B' \subseteq X$ such that $B' \cap A = B$. The connected components of a discrete space being the points, the problem is now solved.
