“Integer broom topology”的意思、由来-开放百科全书

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Integer broom topology

释义

Definition of the integer broom space
Definition of the integer broom topology
Properties
See also
References

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

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 {{nowrap|1=(n,θ) ∈ R²}} such that n is a non-negative integer, and {{nowrap|1=θ ∈ {1/k : k ∈ N and k ≥ 1}}}.^[1] 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 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 a topology on X by means of a product topology. The Integer Broom space is given by the polar coordinates

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.^[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]

References

1. ^¹²{{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}}
2. ^{{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}}

1 : General topology

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Definition of the integer broom space

Definition of the integer broom topology

Properties

See also

References