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:
“With liberty and pleasant weather, the simplest occupation, any unquestioned country mode of life which detains us in the open air, is alluring. The man who picks peas steadily for a living is more than respectable, he is even envied by his shop-worn neighbors. We are as happy as the birds when our Good Genius permits us to pursue any outdoor work, without a sense of dissipation.”
—Henry David Thoreau (1817–1862)
“We know then the existence and nature of the finite, because we also are finite and have extension. We know the existence of the infinite and are ignorant of its nature, because it has extension like us, but not limits like us. But we know neither the existence nor the nature of God, because he has neither extension nor limits.”
—Blaise Pascal (1623–1662)