In
mathematics, the particular point topology (or included point topology) is 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 ho ...
where a
set is
open if it contains a particular point of the
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 po ...
. Formally, let ''X'' be any non-empty set and ''p'' ∈ ''X''. The collection
:
of
subset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...
s of ''X'' is the particular point topology on ''X''. There are a variety of cases that are individually named:
* If ''X'' has two points, the particular point topology on ''X'' is the
Sierpiński space.
* If ''X'' is
finite (with at least 3 points), the topology on ''X'' is called the finite particular point topology.
* If ''X'' is
countably infinite, the topology on ''X'' is called the countable particular point topology.
* If ''X'' is
uncountable, the topology on ''X'' is called the uncountable particular point topology.
A generalization of the particular point topology is the
closed extension topology. In the case when ''X'' \ has the
discrete topology, the closed extension topology is the same as the particular point topology.
This topology is used to provide interesting examples and counterexamples.
Properties
; Closed sets have empty interior
: Given a nonempty open set
every
is a
limit point of ''A''. So the
closure of any open set other than
is
. No
closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric spac ...
other than
contains ''p'' so the
interior
Interior may refer to:
Arts and media
* ''Interior'' (Degas) (also known as ''The Rape''), painting by Edgar Degas
* ''Interior'' (play), 1895 play by Belgian playwright Maurice Maeterlinck
* ''The Interior'' (novel), by Lisa See
* Interior de ...
of every closed set other than
is
.
Connectedness Properties
;Path and locally connected but not
arc connected
For any ''x'', ''y'' ∈ ''X'', the
function ''f'':
, 1→ ''X'' given by
:
is a path. However since ''p'' is open, the
preimage of ''p'' under a
continuous injection from
,1would be an open single point of
,1 which is a contradiction.
;Dispersion point, example of a set with
: ''p'' is a
dispersion point for ''X''. That is ''X'' \ is
totally disconnected.
; Hyperconnected but not ultraconnected
: Every
non-empty open set contains ''p'', and hence ''X'' is
hyperconnected
In the mathematical field of topology, a hyperconnected space or irreducible space is a topological space ''X'' that cannot be written as the union of two proper closed sets (whether disjoint or non-disjoint). The name ''irreducible space'' is pre ...
. But if ''a'' and ''b'' are in ''X'' such that ''p'', ''a'', and ''b'' are three distinct points, then and are
disjoint closed sets and thus ''X'' is not
ultraconnected. Note that if ''X'' is the Sierpiński space then no such ''a'' and ''b'' exist and ''X'' is in fact ultraconnected.
Compactness Properties
; Compact only if finite. Lindelöf only if countable.
: If ''X'' is finite, it is
compact; and if ''X'' is infinite, it is not compact, since the family of all open sets
forms an
open cover with no finite subcover.
: For similar reasons, if ''X'' is countable, it is a
Lindelöf space; and if ''X'' is uncountable, it is not Lindelöf.
; Closure of compact not compact
: The set is compact. However its
closure (the closure of a compact set) is the entire space ''X'', and if ''X'' is infinite this is not compact. For similar reasons if ''X'' is uncountable then we have an example where the closure of a compact set is not a Lindelöf space.
;Pseudocompact but not weakly countably compact
: First there are no disjoint non-empty open sets (since all open sets contain ''p''). Hence every continuous function to the
real line
In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a po ...
must be
constant, and hence bounded, proving that ''X'' is a
pseudocompact space. Any set not containing ''p'' does not have a limit point thus if ''X'' if infinite it is not
weakly countably compact In mathematics, a topological space ''X'' is said to be limit point compact or weakly countably compact if every infinite subset of ''X'' has a limit point in ''X''. This property generalizes a property of compact spaces. In a metric space, limit ...
.
; Locally compact but not locally relatively compact.
: If
, then the set
is a compact
neighborhood of ''x''. However the closure of this neighborhood is all of ''X'', and hence if ''X'' is infinite, ''x'' does not have a closed compact neighborhood, and ''X'' is not
locally relatively 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 ...
.
Limit related
; Accumulation points of sets
: If
does not contain ''p'', ''Y'' has no accumulation point (because ''Y'' is closed in ''X'' and discrete in the subspace topology).
: If
contains ''p'', every point
is an accumulation point of ''Y'', since
(the smallest neighborhood of
) meets ''Y''. ''Y'' has no
ω-accumulation point. Note that ''p'' is never an accumulation point of any set, as it is
isolated
Isolation is the near or complete lack of social contact by an individual.
Isolation or isolated may also refer to:
Sociology and psychology
*Isolation (health care), various measures taken to prevent contagious diseases from being spread
**Is ...
in ''X''.
; Accumulation point as a set but not as a sequence
: Take a sequence
of distinct elements that also contains ''p''. The underlying set
has any
as an accumulation point. However the sequence itself has no
accumulation point as a sequence, as the neighbourhood
of any ''y'' cannot contain infinitely many of the distinct
.
Separation related
; T
0
:''X'' is
T0 (since is open for each ''x'') but satisfies no higher
separation axioms (because all non-empty open sets must contain ''p'').
; Not regular
:Since every non-empty open set contains ''p'', no closed set not containing ''p'' (such as ''X'' \ ) can be
separated by neighbourhoods from , and thus ''X'' is not
regular
The term regular can mean normal or in accordance with rules. It may refer to:
People
* Moses Regular (born 1971), America football player
Arts, entertainment, and media Music
* "Regular" (Badfinger song)
* Regular tunings of stringed instrum ...
. Since
complete regularity implies regularity, ''X'' is not completely regular.
; Not normal
:Since every non-empty open set contains ''p'', no non-empty closed sets can be
separated by neighbourhoods from each other, and thus ''X'' is not
normal. Exception: the
Sierpiński topology is normal, and even completely normal, since it contains no nontrivial separated sets.
Other properties
; Separability
: is
dense and hence ''X'' is a
separable space
In mathematics, a topological space is called separable if it contains a countable, dense subset; that is, there exists a sequence \_^ of elements of the space such that every nonempty open subset of the space contains at least one element o ...
. However if ''X'' is
uncountable then ''X'' \ is not separable. This is an example of a
subspace of a separable space not being separable.
; Countability (first but not second)
: If ''X'' is uncountable then ''X'' is
first countable but not
second countable.
; Alexandrov-discrete
: The topology is an
Alexandrov topology In topology, an Alexandrov topology is a topology in which the intersection of any family of open sets is open. It is an axiom of topology that the intersection of any ''finite'' family of open sets is open; in Alexandrov topologies the finite re ...
. The smallest neighbourhood of a point
is
; Comparable (Homeomorphic topologies on the same set that are not comparable)
: Let
with
. Let
and
. That is ''t''
''q'' is the particular point topology on ''X'' with ''q'' being the distinguished point. Then (''X'',''t''
''p'') and (''X'',''t''
''q'') are
homeomorphic incomparable topologies on the same set.
; No nonempty
dense-in-itself subset
: Let ''S'' be a nonempty subset of ''X''. If ''S'' contains ''p'', then ''p'' is isolated in ''S'' (since it is an isolated point of ''X''). If ''S'' does not contain ''p'', any ''x'' in ''S'' is isolated in ''S''.
; Not first category
: Any set containing ''p'' is dense in ''X''. Hence ''X'' is not a
union of
nowhere dense subsets.
; Subspaces
: Every subspace of a set given the particular point topology that doesn't contain the particular point, has the discrete topology.
See also
*
Alexandrov topology In topology, an Alexandrov topology is a topology in which the intersection of any family of open sets is open. It is an axiom of topology that the intersection of any ''finite'' family of open sets is open; in Alexandrov topologies the finite re ...
*
Excluded point topology
*
Finite topological space
In mathematics, a finite topological space is a topological space for which the underlying set (mathematics), point set is finite set, finite. That is, it is a topological space which has only finitely many elements.
Finite topological spaces are ...
*
List of topologies
*
One-point compactification
*
Overlapping interval topology
References
*{{Citation , last1=Steen , first1=Lynn Arthur , author1-link=Lynn Arthur Steen , last2=Seebach , first2=J. Arthur Jr. , author2-link=J. Arthur Seebach, Jr. , title=
Counterexamples in Topology , origyear=1978 , publisher=
Springer-Verlag , location=Berlin, New York , edition=
Dover
Dover () is a town and major ferry port in Kent, South East England. It faces France across the Strait of Dover, the narrowest part of the English Channel at from Cap Gris Nez in France. It lies south-east of Canterbury and east of Maidstone ...
reprint of 1978 , isbn=978-0-486-68735-3 , mr=507446 , year=1995
Topological spaces