Particular point topology

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This is a topology where inclusion of a particular point defines openness. Let X be any set and $p\in X$ . A non empty subset of X is open if and only if it contains p. There are a variety of cases which are individually named:

If $X = {0,1}$ with $p = 0$ we call X the Sierpiński space.
If X is finite we call the topology on X the Finite Particular Point topology
If X is countable we call the topology on X the Countable Particular Point topology
If X is uncountable we call the topology on X the Uncountable Particular Point topology

A generalization of the particular point topology is the closed extension topology. In the case when $X\backslash \{p\}$ 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.

1 Properties
2 Sierpiński space
3 See also
4 References

[edit] Properties

Empty interior: Every $x\in X$ with $x\ne p$ is a limit point of X. So the closure of any open set other than $\emptyset$ is $X$ . No closed set contains p so the interior of every closed set is $\emptyset$ .

[edit] Connectedness related

Path and locally connected but not arc connected

$f(t) = \begin{cases} x & t\in [0, {1 \over 2}) \\ p & t={1 \over 2} \\ y & t\in ({1 \over 2 }, 1] \end{cases}$

f is path for all x,y ∈ X. However since p is open the inverse image under a continuous one to one 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 open set contains p hence X is hyperconnected. But if $a,b\in X$ with $p\ne a \ne b\, and\, p\ne b$ then {a} and {b} are disjoint closed sets and thus X is not ultraconnected. Note of course that if X is the Sierpinski space then no such a and b exist and X is in fact ultraconnected.

[edit] Compactness related

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 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 ${x, p}$ is a compact neighborhood of x. However the closure of this neighborhood is all of X and hence X 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 $\bigcup_{x\in X} \{p,x\}$ is an open cover with no finite subcover.

[edit] Limit related

Limit point but not an accumulation point: A sequence ${a i}$ converges whenever $\exists x\in X$ such that for all but a finite number of the $a_i,\, a_i \ne p \implies a_i = x.$ An accumulation point will be an $b$ such that infinitely many of the $a i = b$ and note there may be any countable number of b's. Thus any countably infinite set of distinct points forming a sequence does not have an accumulation point but does have a limit point!

Limit point but not a ω-accumulation point: Let Y be any subset containing p. Then for any $q\neq p$ q is a limit point of Y but not a ω-accumulation point. Because this makes no use of properties of Y it leads to often cited counter examples.

[edit] Separation related

T₀: Since there are no disjoing open sets except in the Sierpinski space case X is T₀ but satisfies no higher separation axioms.

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.

Countablility (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 $p,q\in X$ with $p\ne q$ . Let $t_p = \{S\subset X \,|\, p\in S\} \, and\, t_q = \{S\subset X \,|\, q\in S\}$ . 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.

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.

[edit] Sierpiński space

The space $S=(X,\mathcal{T})$ composed of the set $X = {0,1}$ endowed with the topology $\mathcal{T}=\{\varnothing,\{0\},\{0,1\}\}$ is called the connected two-point set or Sierpiński space. It is named after Wacław Sierpiński. It has several properties unique to the two point case (and hence it has its own name):

S is an inaccessible Kolmogorov space; i.e. S satisfies the T₀ axiom, but not the T₁ axiom.
A topological space is Kolmogorov if and only if it is homeomorphic to a subspace of a power of S.
For any topological space X with topology T, let C(X,S) denote the set of all continuous maps from X to S, and for each subset A of X, let I(A) denote the indicator function of A. Then the mapping f : T → C(X,S) defined by f(U) = I(U) is a bijective correspondence.
If X is a topological space with topology T, then the weak topology on X generated by C(X,S) coincides with T.
S is a sober space.
S is generated by the prametric $d (0,1) = 1$ and $d (1,0) = 0$ . That is, S is prametrizable but not metrizable.
See also the article above, particular point topology, which lists many other useful facts about Sierpinski space.