Integer broom topology

In general topology, a branch of mathematics, the integer broom topology is an example of a topology on the so-called integer broom space X.[1]

Definition of the integer broom space

A subset of the integer broom

The integer broom space X is a subset of the plane R². Assume that the plane is parametrised by polar coordinates. The integer broom contains the origin and the points (n, θ) ∈ R² such that n is a non-negative integer and θ ∈ {1/k : k ∈ Z⁺}, where Z⁺ is the set pf positive integers.[1] The image on the right gives an illustration for 0 ≤ n ≤ 5 and 1/15 ≤ θ ≤ 1. Geometrically, the space consists of a collection of convergent sequences. For a fixed n, we have a sequence of points − lying on circle with centre (0, 0) and radius n − that converges to the point (n, 0).

Definition of the integer broom topology

We define the topology on X by means of a product topology. The integer broom space is given by the polar coordinates

${\displaystyle (n,\theta )\in \{n\in \mathbb {Z} :n\geq 0\}\times \{\theta =1/k:k\in \mathbb {Z} ^{+}\}\,.}$

Let us write (n,θ) ∈ U × V for simplicity. The integer broom topology on X is the product topology induced by giving U the right order topology, and V the subspace topology from R.[1]

Properties

The integer broom space, together with the integer broom topology, is a compact topological space. It is a so-called Kolmogorov space, but it is neither a Fréchet space nor a Hausdorff space. The space is path connected, while neither locally connected nor arc connected.[2]

See also

• Comb space
• Infinite broom
• List of topologies

References

1. Steen, L. A.; Seebach, J. A. (1995), Counterexamples in Topology, Dover, p. 140, ISBN 0-486-68735-X
2. Steen, L. A.; Seebach, J. A. (1995), Counterexamples in Topology, Dover, pp. 200–201, ISBN 0-486-68735-X