Particular point topology

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The particular point topology is a topology where sets are considered open if they are empty or contain a particular, arbitrarily chosen, point of the topological space. Formally, let X be any set and p ∈ X. The collection

T = {S ⊆ X: p ∈ S or S = ∅}

of subsets of X is then a particular point topology on X. There are a variety of cases which are individually named:

  • If X = {0,1} we call X the Sierpiński space. This case is somewhat special and is handled separately.
  • If X is finite (with at least 3 points) we call the topology on X the finite particular point topology.
  • If X is countably infinite 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 \ {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.

Contents

[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 other than X contains p so the interior of every closed set other than X is \emptyset .

[edit] Connectedness related

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

[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 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 {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 {ai} 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 ai = 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 non-empty disjoint open sets, X is T0 but satisfies no higher separation axioms.
Not regular
Since every 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.
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 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 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.

[edit] See also

[edit] References