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.

The Sierpiński space has important relations to the theory of computation and semantics ^[1].

[edit] See also

Excluded point topology

[edit] References

^ An online paper, it explains the motivation, why the notion of “topology” can be applied in the investigaton of concepts of the computer science. Alex Simpson: Mathematical Structures for Semantics. Chapter III: Topological Spaces from a Computational Perspective. The “References” section provides many online materials on domain theory.

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Category: Topological spaces