Prove $G$ is connected iff $G$ is connected as a topological space

Question

Let $G$ be a graph. Prove $G$ is connected iff $G$ is connected as a topological space.

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My attempt:

"$\Rightarrow$": Suppose $G$ is connected as a graph. Then all vertices of $G$ are path-connected which in turn implies $G$ as a topological space is path-connected. Then G is also connected.

"$\Leftarrow$": I'm stuck on this implication. I was thinking maybe it would be easier to prove that if $G$ is not connected implies $G$ is not connected as a topological space but no luck so far.

Any help is appreciated!

Answer 1

If is the union of two or more components, then it's the union of pairwise disjoint closed subsets, so your idea works.