Particular point topology


In mathematics, the particular 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
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, 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 \ 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

Connectedness

;Path and locally connected but not arc connected
For any x, y ∈ X, the function f: → X given by
is a path. However, since p is open, the preimage of p under a continuous injection from would be an open single point of , which is a contradiction.
;Dispersion point, example of a set with
; Hyperconnected but not ultraconnected

Compactness

; Compact only if finite. Lindelöf only if countable.
; Closure of compact not compact
;Pseudocompact but not weakly countably compact
; Locally compact but not locally relatively compact.

Limit-related

; Accumulation points of sets
; Accumulation point as a set but not as a sequence

Separation-related

; T0
; Not regular
; Not normal

Other properties

; Separability
; Countability
; Alexandrov-discrete
; Comparable
; No nonempty dense-in-itself subset
; Not first category
; Subspaces