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| 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''.<ref name="CEIT">{{Citation|first=L. A.|last=Steen|first2=J. A.|last2=Seebach|title=[[Counterexamples in Topology]]|publisher=Dover|year=1995|page=140|ISBN=0-486-68735-X}}</ref> To give a set ''X'' a topology means to say which [[subset]]s of ''X'' are [[open set|open]] in a manner that satisfies certain axioms:<ref name="CEIT2">{{Citation|first=L. A.|last=Steen|first2=J. A.|last2=Seebach|title=[[Counterexamples in Topology]]|publisher=Dover|year=1995|page=3|ISBN=0-486-68735-X}}</ref>
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| # The [[union (mathematics)|union]] of open sets is an open set.
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| # The finite [[intersection (mathematics)|intersection]] of open sets is an open set.
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| # ''X'' and the [[empty set]] ∅ are open sets.
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| == Definition of the integer broom space ==
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| [[File:Integer Broom Plot FBN.gif|thumb|<center>A subset of the integer broom</center>]]
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| The integer broom space ''X'' is a subset of the plane '''R'''<sup>2</sup>. Assume that the plane is parametrised by [[polar coordinates]]. The integer broom contains the origin and the points {{nowrap|1=(''n'',θ) ∈ '''R'''<sup>2</sup>}} such that ''n'' is a non-negative [[integer]], and {{nowrap|1=''θ'' ∈ {1/''k'' : ''k'' ∈ '''N''' and ''k'' ≥ 1}}}.<ref name="CEIT"/> The image on the right gives an illustration for {{nowrap|1=0 ≤ ''n'' ≤ 5}} and {{nowrap|1=1/15 ≤ θ ≤ 1}}. Geometrically, the space consists of a series of [[convergent sequence]]s. 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).
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| == Definition of the integer broom topology ==
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| We define a topology on ''X'' by means of a [[product topology]]. The Integer Broom space is given by the polar coordinates
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| :<math> (n,\theta) \in \{ n \in \Z : n \ge 0 \} \times \{ \theta = 1/k : k \in \Z, \ k \ge 1 \} \, . </math>
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| Let us write {{nowrap|1=(''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'''.<ref name="CEIT"/>
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| == Properties ==
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| The integer broom space, together with the integer broom topology, is a [[compact space|compact topological space]]. It is a so-called [[Kolmogorov space]], but it is neither a [[T1 space|Fréchet space]] nor a [[Hausdorff space]]. The space is [[locally connected]] and [[path connected]], while not [[Connected_space#Arc_connectedness|arc connected]].<ref name="CEIT3">{{Citation|first=L. A.|last=Steen|first2=J. A.|last2=Seebach|title=[[Counterexamples in Topology]]|publisher=Dover|year=1995|pages=200 – 201|ISBN=0-486-68735-X}}</ref>
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| == References ==
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| {{reflist}}
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| {{DEFAULTSORT:Integer Broom topology}}
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| [[Category:General topology]]
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