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 p ∈ X. 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
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 .
For any x, y ∈ X, the
functionf: [0, 1] → X given by
is a path. However, since p is open, the
preimage of p under a
continuousinjection from [0,1] would be an open single point of [0,1], which is a contradiction.
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
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.
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 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
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
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.
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
homeomorphicincomparable topologies on the same set.
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.