Niemytzki plane

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In mathematics, the Niemytzki plane (or Nemytskii plane), also sometimes called Nemytskii's tangent disk topology is an example of a topological space; it is a completely regular Hausdorff space (also called Tychonoff space) which is not normal. It is named after Viktor Vladimirovich Nemytskii.

[edit] Construction

The underlying set of the Niemytzki plane is the closed upper half plane $N:=\{(p,q)\in \mathbb R\times \mathbb R: q\ge 0\}$ . A topology is specified by defining neigborhood bases for each point in $N$ :

for each point $(p, q)$ with $q > 0$ we use the Euclidean neighborhoods; that is, the open discs

U p, q (n): = {(x, y):(x - p) 2 + (y - q) 2 < 1 / n 2}

will be used as a neighborhood base.

for each point $(p,0)$ on the x-axis the neighborhood base will consist of discs in the upper half plane that touch the x-axis at the point $(p,0)$ ; that is, we will use the sets

$V_{p}(n):= \{(p,0)\} \cup \{(x,y): (x-p)^2+(y-\frac1n)^2 < 1/n^2\}$

[edit] Proof

The fact that this space N is not normal can be established by the following counting argument:

One the one hand, the countable set $S:=\{(p,q) \in \mathbb Q\times \mathbb Q: q>0\}$ of points with rational coordinates is dense in N; hence every continuous function $f:N\to \mathbb R$ is determined by its restriction to $S$ , so there can be at most $|\mathbb R|^ {|S|} = 2^{\aleph_0}$ many continuous real-valued functions on N.
On the other hand, the real line $L:=\{(p,0): p\in \mathbb R\}$ is a closed discrete subspace of N with $2^{\aleph_0}$ many points. So there are $2^{2^{\aleph_0}} > 2^{\aleph_0}$ many continuous functions from L to $\mathbb R$ . Not all these functions can be extended to continuous functions on N.
Hence N is not normal, because by the Tietze-Urysohn theorem all continuous functions defined on a closed subspace can be extended to a continous function on the whole space.