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In
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 points ...
. 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 many ...
(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 is ...
) that is not
normal Normal(s) or The Normal(s) may refer to: Film and television * ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie * ''Norma ...
. It is named after
Robert Lee Moore Robert Lee Moore (November 14, 1882 – October 4, 1974) was an American mathematician who taught for many years at the University of Texas. He is known for his work in general topology, for the Moore method of teaching university mathematics, ...
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 , doct ...
.


Definition

If \Gamma is the (closed) upper half-plane \Gamma = \, then a
topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
may be defined on \Gamma by taking a
local basis In topology and related areas of mathematics, the neighbourhood system, complete system of neighbourhoods, or neighbourhood filter \mathcal(x) for a point x in a topological space is the collection of all neighbourhoods of x. Definitions Neighbour ...
\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 to ...
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 is ...
), which is not
normal Normal(s) or The Normal(s) may refer to: Film and television * ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie * ''Norma ...
. *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 to ...
, 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 to ...
. Thus, the Moore plane shows that a subspace of a separable space need not be separable. *The Moore plane is
first countable In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space X is said to be first-countable if each point has a countable neighbourhood basis (local base) ...
, but not
second countable In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base. More explicitly, a topological space T is second-countable if there exists some countable collection \mat ...
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 ev ...
. *The Moore plane is
countably metacompact In the mathematical field of general topology, a topological space is said to be metacompact if every open cover has a point-finite open refinement. That is, given any open cover of the topological space, there is a refinement that is again an open ...
but not
metacompact In the mathematical field of general topology, a topological space is said to be metacompact if every open cover has a point-finite open refinement. That is, given any open cover of the topological space, there is a refinement that is again an open ...
.


Proof that the Moore plane is not normal

The fact that this space \Gamma is not
normal Normal(s) or The Normal(s) may refer to: Film and television * ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie * ''Norma ...
can be established by the following counting argument (which is very similar to the argument that the
Sorgenfrey plane In topology, the Sorgenfrey plane is a frequently-cited counterexample to many otherwise plausible-sounding conjectures. It consists of the product of two copies of the Sorgenfrey line, which is the real line \mathbb under the half-open interva ...
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 In topology, the Tietze extension theorem (also known as the Tietze–Urysohn–Brouwer extension theorem) states that continuous functions on a closed subset of a normal topological space can be extended to the entire space, preserving boundedness ...
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) In mathematics, Moore space may refer to: * Moore space (algebraic topology) * Moore space (topology) In mathematics, more specifically point-set topology, a Moore space is a developable regular Hausdorff space. That is, a topological space ''X'' ...
*
Hedgehog space In mathematics, a hedgehog space is a topological space consisting of a set of spines joined at a point. For any cardinal number \kappa, the \kappa-hedgehog space is formed by taking the disjoint union of \kappa real unit intervals identified at t ...


References

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