Particular point topology

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 set and pX. The collection

${\displaystyle T=\{S\subseteq X\mid p\in S{\text{ or }}S=\emptyset \}}$

of subsets of X is the particular point topology on X. There are a variety of cases which 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 an open set ${\displaystyle A\subset X}$  every ${\displaystyle x\neq p}$  is a limit point of A. So the closure of any open set other than ${\displaystyle \emptyset }$  is ${\displaystyle X}$ . No closed set other than ${\displaystyle X}$  contains p so the interior of every closed set other than ${\displaystyle X}$  is ${\displaystyle \emptyset }$ .

Connectedness Properties

Path and locally connected but not arc connected
${\displaystyle f(t)={\begin{cases}x&t=0\\p&t\in (0,1)\\y&t=1\end{cases}}}$
f is a path for all x,yX. 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 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.

Compactness Properties

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 (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.
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 strongly locally compact. Both possibilities regarding global compactness.
If x ∈ X then the set ${\displaystyle \{x,p\}}$  is a compact neighborhood of x. However the closure of this neighborhood is all of X and hence if X is infinite it is not strongly locally 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 ${\displaystyle \bigcup _{x\in X}\{p,x\}}$  is an open cover with no finite subcover.

Limit related

Accumulation point but not a ω-accumulation point
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.
Accumulation point as a set but not as a sequence
Take a sequence {ai} 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 accumulation point y for its neighbourhood {y,p} must contain infinite number of the distinct ai.

Separation related

T0
X is T0 (since {x, p} is open for each x) but satisfies no higher separation axioms (because all open sets must contain p).
Not regular
Since every nonempty 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 nonempty open set contains p, no nonempty closed sets can be separated by neighbourhoods from each other, and thus X is not normal. Exception: the Sierpinski topology is normal, and even completely normal, since it contains no nontrivial separated sets.
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.
Comparable ( Homeomorphic topology on the same set that is not comparable)
Let ${\displaystyle p,q\in X}$  with ${\displaystyle p\neq q}$ . Let ${\displaystyle t_{p}=\{S\subset X\,|\,p\in S\}}$  and ${\displaystyle t_{q}=\{S\subset X\,|\,q\in S\}}$ . 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.
Density (no nonempty subsets dense in themselves)
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 if S is nonempty.
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, inherits the discrete topology.