202303192303

Type :Note Tags :Topology

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R_K

Note

Topology on $\mathbb{R}$ generated by the basis $(a,b)$ and $(a,b)\setminus K$, where $K=\{1/n:n\in \mathbb{N}\}$

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Properties

1. Hausdorff
2. Not Regular. Note that $K$ is closed in ${\mathbb{R}}_K$, we can’t separate $0$ and $K$.

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References

Hausdorff Property Regular Spaces