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Concept
Particular point topology
Summary
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
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.
Closed sets have empty interior
Given a nonempty open set every is a limit point of A. So the closure of any open set other than is . No closed set other than contains p so the interior of every closed set other than is .
Path and locally connected but not arc connected
For any x, y ∈ X, the function f: [0, 1] → X given by
is a path. However since p is open, the 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.
Compact only if finite. Lindelöf only if countable.
Official source
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