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 &isin; 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 $$x \ne p$$ 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,&thinsp;y &isin; 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 $$x\ne p$$ 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 $$x\ne p$$ 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,&thinsp;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 $$p \ne q$$. Let $$t_p = \{S \subseteq X \mid p\in S\}$$ and $$t_q = \{S \subseteq X \mid q\in S\}$$.  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.


 * 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.