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 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
is the (closed) upper half-plane
, 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
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'' 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
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 refers to any completely regular space that i ...
), 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'' 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
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
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 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