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.
Definition of the integer broom space
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 of positive integers. 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
( n , θ ) ∈ { n ∈ Z : n ≥ 0 } × { θ = 1 / k : k ∈ Z + } . {\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.
Properties The integer broom space, together with the integer broom topology, is a compact topological space. It is a T0 space, but it is neither a T1 space nor a Hausdorff space. The space is path connected, while neither locally connected nor arc connected.
See also Comb space Infinite broom List of topologies
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