I was working on a problem that asked me to find an example of a topological space $X$ that satisfies the following:
$X$ is a connected, $X-\{p\}$ is disconnected and $\{p\}$ is open.
I have been unable to find such a space and I have almost convinced myself that there can exist no such space. Any help would be highly appreciated.
The included point topology on a set will work: let $X$ be a set (say infinite) and let $p \in X$ and define the included point topology $$\mathcal{T}_p = \{\emptyset\} \cup \{A \subseteq X: p \in A\}$$
One easily checks this is a topology and $\{p\}$ is open while in $X\setminus \{p\}$ all singletons are relatively open as $q\neq p$ implies $$\{q\} = \{p,q\} \cap (X \setminus \{p\}$$ is open in that subspace, making $X\setminus \{p\}$ infinite discrete and hence disconnected.
$X$ itself is connected as any two non-empty open sets must intersect in $p$, so there can be decomposition of $X$ into two non-empty disjoint open sets.