Broom Space


The subset B of the Euclidean plane formed by the union of the interval [0,1] of the x-axis and all line segments of unit length passing through the origin which form an angle alpha=1/n (measured in radians) with it, for all positive integers n.

With respect to the relative topology, B is pathwise-connected. Therefore it is connected, but it is not locally pathwise-connected at any point of the open interval (0,1). Each disk centered at one of these points intersects B in a union of disjoint segments, which form a disconnected set.


Let B_n be the broom space formed by segments of length 1/2^n for all natural numbers n, and place B_0, B_1, B_2, ... one right after the other on the x-axis. This will cover the half-open interval [0,2) of the x-axis (above figure). The space obtained by adding the point (2,0) to this sequence of brooms is then connected im kleinen at point (2,0), since each open neighborhood of (2,0) contains a closed disk whose radius is exactly formed by the basis intervals of B_n for all sufficiently large n. Hence any two points contained in this disk are connected by a path composed of segments of these broom spaces. On the other hand, point (2, 0) has no connected open neighborhood since every open disk centered at (2,0) has no boundary and hence, unlike in the case of a closed disk, it cannot end right at the vertex of some broom space. Therefore, it must cut through some B_n, which will be intersected in an union of disjoint segments, and these form a disconnected set.

See also

Comb Space

This entry contributed by Margherita Barile

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Joshi, K. D. Introduction to General Topology. New Delhi, India: Wiley, p. 157, 1983.

Referenced on Wolfram|Alpha

Broom Space

Cite this as:

Barile, Margherita. "Broom Space." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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