# Homeomorphic characterization of the real line? [duplicate]

Let $$A$$ be a path-connected subset of $$\mathbb R^2$$ such that the removal of any singleton from $$A$$ splits $$A$$ into two open connected components, each of which is path-connected.

Is $$A$$ necessarily homeomorphic to $$\mathbb{R}$$?

## marked as duplicate by Lee Mosher, Chris Godsil, Ben McKay, Community♦Jan 8 at 16:59

• In order to apply this result, we need to establish the local connectednes sof $A$. And how to prove this fact? – Taras Banakh Jan 5 at 21:35