Niemytzki plane
Let Γ be the Euclidean half plane Γ={(x,y)∣y≥0}⊆ℝ2, with the usual subspace topology. We enrich the topology on Γ by throwing in open sets of the form {(x,0)}∪Br(x,r), that is an open ball of radius r around (x,r) together with its point tangent to ℝ×{0} (Fig. 1).
The space Γ endowed with the enriched topology is called the Niemytzki plane.
Some miscellaneous properties of the Niemytzki plane are
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the subspace ℝ×{0} of Γ is discrete, hence the only convergent sequences in this subspace are constant ones;
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it is Hausdorff;
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it is completely regular;
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it is not normal.
Title | Niemytzki plane |
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Canonical name | NiemytzkiPlane |
Date of creation | 2013-03-22 13:36:53 |
Last modified on | 2013-03-22 13:36:53 |
Owner | PrimeFan (13766) |
Last modified by | PrimeFan (13766) |
Numerical id | 7 |
Author | PrimeFan (13766) |
Entry type | Example |
Classification | msc 54-00 |
Classification | msc 54G99 |
Synonym | Niemytzki space |