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 non-empty set and p ∈ X. The collection

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 $A\subseteq X$ every $x\neq p$ is a limit point of A. 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 $\emptyset$ .

Connectedness Properties

Path and locally connected but not arc connected

For any x, y ∈ X, the function f: [0, 1] → X given by

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 Properties

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 $\{p,x\}\;(x\in X)$ 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 $x\in X$ , then the set $\{x,p\}$ 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.

Limit related

Accumulation points of sets: If $Y\subseteq X$ does not contain p, Y has no accumulation point (because Y is closed in X and discrete in the subspace topology).

If

Y\subseteq X

contains p, every point

x\neq p

is an accumulation point of Y, since

\{x,p\}

(the smallest neighborhood of

x

) meets Y. Y has no ω-accumulation point. Note that p is never an accumulation point of any set, as it is isolated in X.

Accumulation point as a set but not as a sequence: Take a sequence $(a_{n})_{n}$ of distinct elements that also contains p. The underlying set $\{a_{n}\}$ has any $x\neq p$ as an accumulation point. However the sequence itself has no accumulation point as a sequence, as the neighbourhood $\{y,p\}$ of any y cannot contain infinitely many of the distinct $a_{n}$ .

Separation related

T₀: X is T₀ (since {x, p} is open for each x) but satisfies no higher separation axioms (because all non-empty open sets must contain p).

Not regular: Since every non-empty 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 non-empty open set contains p, no non-empty closed sets can be separated by neighbourhoods from each other, and thus X is not normal. Exception: the Sierpiński topology is normal, and even completely normal, since it contains no nontrivial separated sets.

Other properties

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.

Alexandrov-discrete: The topology is an Alexandrov topology. The smallest neighbourhood of a point $x$ is $\{x,p\}.$

Comparable (Homeomorphic topologies on the same set that are not comparable): Let $p,q\in X$ with $p\neq q$ . Let $t_{p}=\{S\subseteq X\mid p\in S\}$ and $t_{q}=\{S\subseteq X\mid q\in S\}$ . 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.

No nonempty dense-in-itself subset: Let S be a nonempty subset of X. If S contains p, then p is isolated in S (since it is an isolated point of X). If S does not contain p, any x in S is isolated in S.

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, has the discrete topology.

References

Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978], Counterexamples in Topology (Dover reprint of 1978 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-486-68735-3, MR 0507446