In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, 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 Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
. It is a
completely regular
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 is any completely regular space that is also a ...
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space ( , ), T2 space or separated space, is a topological space where distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topologi ...
(that is, a
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 is any completely regular space that is also a ...
) that is not
normal. It is an example of a
Moore space that is not
metrizable
In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \tau) is said to be metrizable if there is a metric d : X \times X \to , \infty) suc ...
. 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
Viktor Vladimirovich Nemytskii (; 22 November 1900 – 7 August 1967) was a Soviet Union, Soviet mathematician who introduced Nemytskii operators and the Nemytskii plane (Moore plane). He was married to Nina Bari, who was also a mathematician.
Wo ...
.
Definition
If
is the (closed) upper half-plane
, then a
topology
Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
may be defined on
by taking a
local basis as follows:
*Elements of the local basis at points
with
are the open discs in the plane which are small enough to lie within
.
*Elements of the local basis at points
are sets
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
:
Thus the
subspace topology
In topology and related areas of mathematics, a subspace of a topological space (''X'', ''𝜏'') is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''𝜏'' called the subspace topology (or the relative topology ...
inherited by
is the same as the subspace topology inherited from the standard topology of the Euclidean plane.
Properties
*The Moore plane
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 is any completely regular space that is also a ...
), which is not
normal.
*The subspace
of
has, as its
subspace topology
In topology and related areas of mathematics, a subspace of a topological space (''X'', ''𝜏'') is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''𝜏'' called the subspace topology (or the relative topology ...
, 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, 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 e ...
.
*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 op ...
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 op ...
.
Proof that the Moore plane is not normal
The fact that this space
is not
normal 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 inter ...
is not normal):
# On the one hand, the countable set
of points with rational coordinates is dense in
; hence every continuous function
is determined by its restriction to
, so there can be at most
many continuous real-valued functions on
.
# On the other hand, the real line
is a closed discrete subspace of
with
many points. So there are
many continuous functions from ''L'' to
. Not all these functions can be extended to continuous functions on
.
# Hence
is not normal, because by the
Tietze extension theorem
In topology, the Tietze extension theorem (also known as the Tietze– Urysohn– Brouwer extension theorem or Urysohn-Brouwer lemma) states that any real-valued, continuous function on a closed subset of a normal topological space
In mathe ...
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
*
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 ...
References
* Stephen Willard. ''General Topology'', (1970) Addison-Wesley .
* ''(Example 82)''
* {{planetmathref, urlname=NiemytzkiPlane, title= Niemytzki plane
Topological spaces