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
T = { S ⊆ X ∣ p ∈ S } ∪ { ∅ } {\displaystyle T=\{S\subseteq X\mid p\in S\}\cup \{\emptyset \}}
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 A ⊆ X {\displaystyle A\subseteq X} every x ≠ p {\displaystyle x\neq p} is a limit point of A. So the closure of any open set other than ∅ {\displaystyle \emptyset } is X {\displaystyle X} . No closed set other than X {\displaystyle X} contains p so the interior of every closed set other than X {\displaystyle X} is ∅ {\displaystyle \emptyset } .
Connectedness Path and locally connected but not arc connected For any x, y ∈ X, the function f: [0, 1] → X given by
f ( t ) = { x t = 0 p t ∈ ( 0 , 1 ) y t = 1 {\displaystyle f(t)={\begin{cases}x&t=0\\p&t\in (0,1)\\y&t=1\end{cases}}}
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 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 { p , x } ( x ∈ X ) {\displaystyle \{p,x\}\;(x\in X)} 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 x ∈ X {\displaystyle x\in X} , then the set { x , p } {\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, x does not have a closed compact neighborhood, and X is not locally relatively compact.
… excerpt ends here. Continue reading the full article.
