Examples of topology that results in a connected topological space when $X = \{1,2,3\}$

Just starting off a chapter on Connectedness in Adams Introduction to Topology: Pure and Applied and I'm looking to get a concrete example .. The book states, "Let $X$ be a topological space. We call $X$ connected if there does not exist a pair of disjoint nonempty open sets whose union is $X$."

So say we are given $X=\{1,2,3\}$, then the follow would result in a connected topological space: $T=\{X,\emptyset,\{1\},\{3\},\{1,3\}\}$, $T=\{X,\emptyset,\{1\},\{1,3\}\}$, and $T=\{X,\emptyset,\{2\}\}$

Alternative, $T=\{X,\emptyset,\{1\},\{2\},\{1,3\}\}$ would result in a disconnected topological space because the union of $\{2\}$ and $\{1,3\}$ is $X$, and they are disjoint. Same logic applies to $T=\{X,\emptyset,\{1\},\{2,3\}\}$.

Is this example correct- with correct application of the definition? Thanks.

The first three examples work fine. Note that your fourth example $T = \{X, \emptyset, \{1\}, \{2\}, \{1, 3\}\}$ is not a topology, since the union of the open sets $\{1\}$ and $\{2\}$ is not open. Your fifth example does work however.