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In [[mathematics]], the '''particular point topology''' (or '''included point topology''') is a [[topological space|topology]] where [[set (mathematics)|set]]s are considered [[open set|open]] if they are empty or contain a particular, arbitrarily chosen, point of the [[topological space]]. Formally, let ''X'' be any set and ''p'' ∈ ''X''. The collection
In [[mathematics]], the '''particular point topology''' (or '''included point topology''') is a [[topological space|topology]] where a [[set (mathematics)|set]] is [[open set|open]] if it contains a particular point of the [[topological space]]. Formally, let ''X'' be any non-empty set and ''p'' ∈ ''X''. The collection
:''T'' = {''S'' ⊆ ''X'': ''p'' ∈ ''S'' or ''S'' = ∅}
:<math>T = \{S \subseteq X \mid p \in S \} \cup \{\emptyset\}</math>
of subsets of ''X'' is then the particular point topology on ''X''. There are a variety of cases which are individually named:
of [[subset]]s of ''X'' is the particular point topology on ''X''. There are a variety of cases that are individually named:


* If ''X'' = {0,1} we call ''X'' the '''[[Sierpiński space]]'''. This case is somewhat special and is handled separately.
* If ''X'' has two points, the particular point topology on ''X'' is the [[Sierpiński space]].
* If ''X'' is [[finite set|finite]] (with at least 3 points) we call the topology on ''X'' the '''finite particular point topology'''.
* If ''X'' is [[finite set|finite]] (with at least 3 points), the topology on ''X'' is called the '''finite particular point topology'''.
* If ''X'' is [[countably infinite]] we call the topology on ''X'' the '''countable particular point topology'''.
* If ''X'' is [[countably infinite]], the topology on ''X'' is called the '''countable particular point topology'''.
* If ''X'' is [[uncountable]] we call the topology on ''X'' the '''uncountable 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'' \ {''p''} has the discrete topology, the closed extension topology is the same as the particular point topology.
A generalization of the particular point topology is the [[closed extension topology]]. In the case when ''X'' \ {''p''} 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.
This topology is used to provide interesting examples and counterexamples.
Line 14: Line 14:
==Properties==
==Properties==
; Closed sets have empty interior
; Closed sets have empty interior
: Given an open set <math>A \subset X</math> every <math>x \ne p</math> is a [[limit point]] of A. So the closure of any open set other than <math>\emptyset</math> is <math>X</math>. No closed set other than <math>X</math> contains p so the interior of every closed set other than <math>X</math> is <math>\emptyset</math>.
: Given a nonempty open set <math>A \subseteq X</math> every <math>x \ne p</math> is a [[limit point]] of ''A''. So the [[Closure (topology)|closure]] of any open set other than <math>\emptyset</math> is <math>X</math>. No [[closed set]] other than <math>X</math> contains ''p'' so the [[Interior (topology)|interior]] of every closed set other than <math>X</math> is <math>\emptyset</math>.


===Connectedness Properties===
===Connectedness Properties===
;Path and locally connected but not [[connected space|arc connected]]
;Path and locally connected but not [[connected space#Arc connectedness|arc connected]]


For any ''x'',&thinsp;''y'' &isin; ''X'', the [[Function (mathematics)|function]] ''f'': [0, 1] → ''X'' given by
: <math>
f(t) = \begin{cases} x & t=0 \\
: <math>f(t) = \begin{cases} x & t=0 \\
p & t\in(0,1) \\
p & t\in(0,1) \\
y & t=1
y & t=1
\end{cases}</math>
\end{cases}</math>


: ''f'' is a path for all ''x'',''y'' &isin; ''X''. However since ''p'' is open, the preimage of ''p'' under a continuous injection from [0,1] would be an open single point of [0,1], which is a contradiction.
is a path. However, since ''p'' is open, the [[preimage]] of ''p'' under a [[Continuous function (topology)|continuous]] [[Injective|injection]] from [0,1] would be an open single point of [0,1], which is a contradiction.


;Dispersion point, example of a set with
;Dispersion point, example of a set with
: ''p'' is a '''[[dispersion point]]''' for ''X''. That is ''X\{p}'' is [[totally disconnected]].
: ''p'' is a '''[[dispersion point]]''' for ''X''. That is ''X'' \ {''p''} is [[totally disconnected]].


; Hyperconnected but not ultraconnected
; Hyperconnected but not ultraconnected
: Every open set contains ''p'' hence ''X'' is [[hyperconnected]]. But if ''a'' and ''b'' are in ''X'' such that ''p'', ''a'', and ''b'' are three distinct points, then ''{a}'' and ''{b}'' are disjoint closed sets and thus ''X'' is not [[ultraconnected]]. Note that if ''X'' is the Sierpinski space then no such ''a'' and ''b'' exist and ''X'' is in fact ultraconnected.
: Every [[Empty set|non-empty]] open set contains ''p'', and hence ''X'' is [[hyperconnected]]. But if ''a'' and ''b'' are in ''X'' such that ''p'', ''a'', and ''b'' are three distinct points, then {''a''} and {''b''} are [[Disjoint sets|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===
===Compactness Properties===
; Compact only if finite. Lindelöf only if countable.
: If ''X'' is finite, it is [[Compact space|compact]]; and if ''X'' is infinite, it is not compact, since the family of all open sets <math>\{p,x\}\;(x\in X)</math> forms an [[Cover (topology)#Cover in topology|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
; Closure of compact not compact
: The set ''{p}'' is [[Compact space|compact]]. However its [[Closure (topology)|closure]] (the closure of a compact set) is the entire space ''X'' and if ''X'' is infinite this is not compact (since any set ''{t,p}'' is open). For similar reasons if ''X'' is uncountable then we have an example where the closure of a compact set is not a [[Lindelöf space]].
: The set {''p''} is compact. However its [[Closure (topology)|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
;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]] 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]].
: First there are no disjoint non-empty open sets (since all open sets contain ''p''). Hence every continuous function to the [[real line]] must be [[Constant function|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]].


; Locally compact but not strongly locally compact. Both possibilities regarding global compactness.
; Locally compact but not locally relatively compact.
: If ''x &isin; X'' then the set <math>\{x,p\}</math> is a compact neighborhood of ''x''. However the closure of this neighborhood is all of ''X'' and hence ''X'' is not strongly locally compact.
: If <math>x\in X</math>, then the set <math>\{x,p\}</math> is a compact [[Neighborhood (mathematics)|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 terms of global compactness, ''X'' finite if and only if ''X'' is compact. The first implication is immediate, the reverse implication follows from noting that <math>\bigcup_{x\in X} \{p,x\} </math> is an open cover with no finite subcover.


===Limit related===
===Limit related===
; Accumulation point but not a &omega;-accumulation point
; Accumulation points of sets
: If <math>Y\subseteq X</math> does not contain ''p'', ''Y'' has no accumulation point (because ''Y'' is closed in ''X'' and discrete in the subspace topology).
: If ''Y'' is some subset containing ''p'' then any ''x'' different from ''p'' is an accumulation point of ''Y''. However ''x'' is not an ''[[ω-accumulation point]]'' as {''x'',''p''} is one neighbourhood of ''x'' which does not contain infinitely many points from ''Y''. Because this makes no use of properties of ''Y'' it leads to often cited counter examples.

: If <math>Y\subseteq X</math> contains ''p'', every point <math>x\ne p</math> is an accumulation point of ''Y'', since <math>\{x,p\}</math> (the smallest neighborhood of <math>x</math>) meets ''Y''. ''Y'' has no [[ω-accumulation point]]. Note that ''p'' is never an accumulation point of any set, as it is [[isolated point|isolated]] in ''X''.


; Accumulation point as a set but not as a sequence
; Accumulation point as a set but not as a sequence
: Take a sequence {''a''<sub>''i''</sub>} of distinct elements that also contains ''p''. As in the example above, the underlying set has any ''x'' different from ''p'' as an accumulation point. However the sequence itself cannot possess an accumulation point ''y'' for its neighbourhood {''y'', ''p''} must contain infinite number of the distinct ''a''<sub>''i''</sub>.
: Take a sequence <math>(a_n)_n</math> of distinct elements that also contains ''p''. The underlying set <math>\{a_n\}</math> has any <math>x\ne p</math> as an accumulation point. However the sequence itself has no [[accumulation point#For sequences and nets|accumulation point as a sequence]], as the neighbourhood <math>\{y,p\}</math> of any ''y'' cannot contain infinitely many of the distinct <math>a_n</math>.


===Separation related===
===Separation related===


; T<sub>0</sub>
; T<sub>0</sub>
:''X'' is [[Kolmogorov space|T<sub>0</sub>]] (since {''x'', ''p''} is open for each ''x'') but satisfies no higher [[separation axiom]]s (because all open sets must contain ''p'').
:''X'' is [[Kolmogorov space|T<sub>0</sub>]] (since {''x'',&thinsp;''p''} is open for each ''x'') but satisfies no higher [[separation axiom]]s (because all non-empty open sets must contain ''p'').


; Not regular
; Not regular
:Since every nonempty open set contains ''p'', no closed set not containing ''p'' (such as ''X''\{''p''}) can be [[separated set|separated by neighbourhoods]] from {''p''}, and thus ''X'' is not [[regular space|regular]]. Since [[tychonoff space|complete regularity]] implies regularity, ''X'' is not completely regular.
:Since every non-empty open set contains ''p'', no closed set not containing ''p'' (such as ''X'' \ {''p''}) can be [[separated set|separated by neighbourhoods]] from {''p''}, and thus ''X'' is not [[Regular space|regular]]. Since [[Tychonoff space|complete regularity]] implies regularity, ''X'' is not completely regular.


; Not normal
; Not normal
:Since every nonempty open set contains ''p'', no nonempty closed sets can be [[separated set|separated by neighbourhoods]] from each other, and thus ''X'' is not [[normal space|normal]]. Exception: the [[Sierpinski topology]] is normal, and even completely normal, since it contains no nontrivial separated sets.
:Since every non-empty open set contains ''p'', no non-empty closed sets can be [[separated set|separated by neighbourhoods]] from each other, and thus ''X'' is not [[normal space|normal]]. Exception: the [[Sierpinski topology|Sierpiński topology]] is normal, and even completely normal, since it contains no nontrivial separated sets.


===Other properties===
;Separability

: ''{p}'' is dense and hence ''X'' is a [[separable space]]. However if ''X'' is [[Uncountable set|uncountable]] then ''X\{p}'' is not separable. This is an example of a subspace of a separable space not being separable.
; Separability
: {''p''} is [[Dense set|dense]] and hence ''X'' is a [[separable space]]. However if ''X'' is [[Uncountable set|uncountable]] then ''X'' \ {''p''} is not separable. This is an example of a [[Subspace topology|subspace]] of a separable space not being separable.


; Countability (first but not second)
; Countability (first but not second)
: If ''X'' is uncountable then ''X'' is [[First-countable space|first countable]] but not [[Second-countable space|second countable]].
: If ''X'' is uncountable then ''X'' is [[First-countable space|first countable]] but not [[Second-countable space|second countable]].


; Alexandrov-discrete
; Comparable ( Homeomorphic topology on the same set that is not comparable)
: The topology is an [[Alexandrov topology]]. The smallest neighbourhood of a point <math>x</math> is <math>\{x,p\}.</math>
: Let <math> p,q\in X</math> with <math>p\ne q</math>. Let <math>t_p = \{S\subset X \,|\, p\in S\}</math> and <math>t_q = \{S\subset X \,|\, q\in S\}</math>. That is ''t<sub>q</sub>'' is the particular point topology on ''X'' with ''q'' being the distinguished point. Then ''(X,t<sub>p</sub>)'' and ''(X,t<sub>q</sub>)'' are homeomorphic [[Comparison of topologies|incomparable topologies]] on the same set.


; Comparable (Homeomorphic topologies on the same set that are not comparable)
; Density (no nonempty subsets dense in themselves)
: Let <math>p, q \in X</math> with <math>p \ne q</math>. Let <math>t_p = \{S \subseteq X \mid p\in S\}</math> and <math>t_q = \{S \subseteq X \mid q\in S\}</math>. That is ''t''<sub>''q''</sub> is the particular point topology on ''X'' with ''q'' being the distinguished point. Then (''X'',''t''<sub>''p''</sub>) and (''X'',''t''<sub>''q''</sub>) are [[homeomorphic]] [[Comparison of topologies|incomparable topologies]] on the same set.
: Let ''S'' be a subset of ''X''. If ''S'' contains ''p'' then S has no limit points (see limit point section). If ''S'' does not contain ''p'' then ''p'' is not a limit point of ''S''. Hence ''S'' is not [[Dense set|dense]] if ''S'' is nonempty.


; No nonempty [[dense-in-itself]] subset
;Not first category
: 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''.
: Any set containing ''p'' is dense in ''X''. Hence ''X'' is not a union of [[Baire space|nowhere dense subsets]].


; Not first category
;Subspaces
: Any set containing ''p'' is dense in ''X''. Hence ''X'' is not a [[Union (set theory)|union]] of [[Baire space|nowhere dense subsets]].
: Every subspace of a set given the particular point topology that doesn't contain the particular point, inherits the discrete topology.

; Subspaces
: Every subspace of a set given the particular point topology that doesn't contain the particular point, has the discrete topology.

== See also ==


==See also==
* [[Sierpiński space]]
* [[Excluded point topology]]
* [[Alexandrov topology]]
* [[Alexandrov topology]]
* [[Excluded point topology]]
* [[Finite topological space]]
* [[Finite topological space]]
* [[List of topologies]]
* [[One-point compactification]]
* [[One-point compactification]]
* [[Overlapping interval topology]]


==References==
==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 Publications|Dover]] reprint of 1978 | isbn=978-0-486-68735-3 | id={{MathSciNet|id=507446}} | year=1995}}
*{{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 Publications|Dover]] reprint of 1978 | isbn=978-0-486-68735-3 |mr=507446 | year=1995}}


[[Category:Topological spaces]]
[[Category:Topological spaces]]

Latest revision as of 21:23, 24 August 2023

In mathematics, the particular point topology (or included point topology) is a topology where a set is open if it contains a particular point of the topological space. Formally, let X be any non-empty set and pX. The collection

of subsets 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 \ {p} 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

[edit]
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 other than contains p so the interior of every closed set other than is .

Connectedness Properties

[edit]
Path and locally connected but not arc connected

For any x, yX, the function f: [0, 1] → X given by

is a path. However, since p is open, the preimage of p under a continuous injection from [0,1] would be an open single point of [0,1], which is a contradiction.

Dispersion point, example of a set with
p is a dispersion point for X. That is X \ {p} is totally disconnected.
Hyperconnected but not ultraconnected
Every non-empty open set contains p, and hence X is hyperconnected. But if a and b are in X such that p, a, and b are three distinct points, then {a} and {b} 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

[edit]
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 {p} 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 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.
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.
[edit]
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 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 .
[edit]
T0
X is T0 (since {x, p} 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 \ {p}) can be separated by neighbourhoods from {p}, and thus X is not regular. 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

[edit]
Separability
{p} is dense and hence X is a separable space. However if X is uncountable then X \ {p} 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. The smallest neighbourhood of a point is
Comparable (Homeomorphic topologies on the same set that are not comparable)
Let with . Let and . That is tq is the particular point topology on X with q being the distinguished point. Then (X,tp) and (X,tq) 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

[edit]

References

[edit]
  • Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978], Counterexamples in Topology (Dover reprint of 1978 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-486-68735-3, MR 0507446