Points are Path-Connected iff Contained in Path-Connected Set

Theorem
Let $T = \left({S, \tau}\right)$ be a topological space.

Let $x, y \in S$

Then:
 * $x, y$ are path-connected points in $T$ there exists a path-connected set of $T$ containing $x$ and $y$.

Necessary Condition
Let $x, y$ be path-connected points in $T$.

Let $\gamma : \closedint 0 1 \to T$ be a path from $x$ to $y$.

From Image of Path is Path-Connected Set, $\map \gamma {\closedint 0 1}$ is a path-connected set of $T$ containing $x$ and $y$.

The result follows.

Sufficient Condition
Let $B$ be a path-connected set of $T$ containing $x$ and $y$.

Then there exists a path $\gamma : \closedint 0 1 \to B$ from $x$ to $y$.

Let $i_B : B \to S$ be the inclusion mapping from $B$ into $S$.

From Composite of Continuous Mappings is Continuous, $i_B \circ g : \closedint 0 1 \to T$ is continuous.

Hence $i_B \circ g$ is a path from $x$ to $y$ in $T$.