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 set and p ∈ X. The collection
of subsets of X is the particular point topology on X. There are a variety of cases which are individually named:
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 Properties

;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 Properties

; Closure of compact not compact
;Pseudocompact but not weakly countably compact
; Locally compact but not strongly locally compact. Both possibilities regarding global compactness.

Limit related

; Accumulation point but not a ω-accumulation point
; Accumulation point as a set but not as a sequence

Separation related

; T0
; Not regular
; Not normal
;Separability
; Countability
; Comparable
; Density
;Not first category
;Subspaces