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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
of subsets of X is the particular point topology on X. There are a variety of cases that are individually named:
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.
For any x, y ∈ X, the function f: [0, 1] → X given by
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.
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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
of subsets of X is the particular point topology on X. There are a variety of cases that are individually named:
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.
For any x, y ∈ X, the function f: [0, 1] → X given by
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.