Open Extension Topology
Let X be a topological space and P a set disjoint from X. Consider in X ∪ P the topology whose open sets are of the form: X ∪ Q, where Q is a subset of P, or A, where A is an open set of X.
For this reason this topology is called the open extension topology of X plus P, with which one extends to X ∪ P the open sets of X. Note that the subspace topology of X as a subset of X ∪ P is the original topology of X, while the subspace topology of P as a subset of X ∪ P is the discrete topology.
Note that the closed sets of X ∪ P are of the form: Q, where Q is a subset of P, or B ∪ P, where B is a closed set of X.
Being Y a topological space and R a subset of Y, one might ask whether the extension topology of Y - R plus R is the same as the original topology of Y, and the answer is in general no.
Note that the open extension topology of X ∪ P is smaller than the extension topology of X ∪ P.
Being Z a set and p a point in Z, one obtains the excluded point topology construction by considering in Z the discrete topology and applying the open extension topology construction to Z - {p} plus p.
Read more about this topic: Extension Topology
Famous quotes containing the words open and/or extension:
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Shalt constant faith and service vow,
Thy Celia shall receive those charms
With open ears, and with unfolded arms.”
—Thomas Carew (15891639)
“The motive of science was the extension of man, on all sides, into Nature, till his hands should touch the stars, his eyes see through the earth, his ears understand the language of beast and bird, and the sense of the wind; and, through his sympathy, heaven and earth should talk with him. But that is not our science.”
—Ralph Waldo Emerson (18031882)