Show that the property of having a cut point is a topological property

Question

We know a point p of a topological space is a cut point if X\{p} is disconnected.

Answer 1

Suppose f : X -> Y is a homeomorphism from a topological space X having a cut point, p. Let q = f(p). The restriction of f to X\{p} is a homeomorphism X \ {p} ≈ Y \ {q}; since X \ {p} is disconnected then Y \ {q} is disconnected. Therefore q is a cut point of Y.