Infinite broom

In topology, a branch of mathematics, the infinite broom is a subset of the Euclidean plane that is used as an example distinguishing various notions of connectedness.

[1] The infinite broom is the subset of the Euclidean plane that consists of all closed line segments joining the origin to the point (1, 1/n) as n varies over all positive integers, together with the interval (½, 1] on the x-axis.

[2] Both the infinite broom and its closure are connected, as every open set in the plane which contains the segment on the x-axis must intersect slanted segments.

Despite the closed infinite broom being arc connected, the standard infinite broom is not path connected.

[2] The interval [0,1] on the x-axis is a deformation retract of the closed infinite broom, but it is not a strong deformation retract.