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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Moore plane, also sometimes called Niemytzki plane (or Nemytskii plane, Nemytskii's tangent disk topology), is a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poi ...
. It is a completely regular
Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the man ...
(also called
Tychonoff space In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are kinds of topological spaces. These conditions are examples of separation axioms. A Tychonoff space refers to any completely regular space that i ...
) that is not normal. It is named after Robert Lee Moore and
Viktor Vladimirovich Nemytskii , birth_date = , birth_place = Smolensk , citizenship = Soviet Union , nationality = , death_date = , death_place = Sayan Mountains , field = Mathematics , work_institution = Moscow State University , alma_mater = Moscow State University , ...
.


Definition

If \Gamma is the (closed) upper half-plane \Gamma = \, then a
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
may be defined on \Gamma by taking a local basis \mathcal(p,q) as follows: *Elements of the local basis at points (x,y) with y>0 are the open discs in the plane which are small enough to lie within \Gamma. *Elements of the local basis at points p = (x,0) are sets \\cup A where ''A'' is an open disc in the upper half-plane which is tangent to the ''x'' axis at ''p''. That is, the local basis is given by :\mathcal(p,q) = \begin \, & \mbox q > 0; \\ \, & \mbox q = 0. \end Thus the
subspace topology In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced t ...
inherited by \Gamma\backslash \ is the same as the subspace topology inherited from the standard topology of the Euclidean plane.


Properties

*The Moore plane \Gamma is separable, that is, it has a countable dense subset. *The Moore plane is a completely regular Hausdorff space (i.e.
Tychonoff space In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are kinds of topological spaces. These conditions are examples of separation axioms. A Tychonoff space refers to any completely regular space that i ...
), which is not normal. *The subspace \ of \Gamma has, as its
subspace topology In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced t ...
, the
discrete topology In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest t ...
. Thus, the Moore plane shows that a subspace of a separable space need not be separable. *The Moore plane is first countable, but not second countable or Lindelöf. *The Moore plane is not
locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ...
. *The Moore plane is countably metacompact but not metacompact.


Proof that the Moore plane is not normal

The fact that this space \Gamma is not normal can be established by the following counting argument (which is very similar to the argument that the Sorgenfrey plane is not normal): # On the one hand, the countable set S:=\ of points with rational coordinates is dense in \Gamma; hence every continuous function f:\Gamma \to \mathbb R is determined by its restriction to S, so there can be at most , \mathbb R, ^ = 2^ many continuous real-valued functions on \Gamma. # On the other hand, the real line L:=\ is a closed discrete subspace of \Gamma with 2^ many points. So there are 2^ > 2^ many continuous functions from ''L'' to \mathbb R. Not all these functions can be extended to continuous functions on \Gamma. # Hence \Gamma is not normal, because by the Tietze extension theorem all continuous functions defined on a closed subspace of a normal space can be extended to a continuous function on the whole space. In fact, if ''X'' is a separable topological space having an uncountable closed discrete subspace, ''X'' cannot be normal.


See also

* Moore space (disambiguation) * Hedgehog space


References

* Stephen Willard. ''General Topology'', (1970) Addison-Wesley . * ''(Example 82)'' * {{planetmathref, urlname=NiemytzkiPlane, title= Niemytzki plane Topological spaces