Extension topology

In topology, a branch of mathematics, an extension topology is a topology placed on the disjoint union of a topological space and another set.
There are various types of extension topology, described in the sections below.
Contents
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: A ∪ Q, where A is an open set of X and Q is a subset of P.
Note that the closed sets of X ∪ P are of the form: B ∪ Q, where B is a closed set of X and Q is a subset of P.
For these reasons this topology is called the extension topology of X plus P, with which one extends to X ∪ P the open and the closed 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.
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 the similitude of this extension topology construction and the Alexandroff onepoint compactification, in which case, having a topological space X which one wishes to compactify by adding a point ∞ in infinity, one considers the closed sets of X ∪ {∞} to be the sets of the form: K, where K is a closed compact set of X, or B ∪ {∞}, where B is a closed set of X.
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.
Closed extension topology
Let X be a topological space and P a set disjoint from X. Consider in X ∪ P the topology whose closed sets are of the form: X ∪ Q, where Q is a subset of P, or B, where B is a closed set of X.
For this reason this topology is called the closed extension topology of X plus P, with which one extends to X ∪ P the closed 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 open sets of X ∪ P are of the form: Q, where Q is a subset of P, or A ∪ P, where A is an open 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 closed 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 particular point topology construction by considering in Z the discrete topology and applying the closed extension topology construction to Z  {p} plus p.
References
 Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978], Counterexamples in Topology (Dover reprint of 1978 ed.), Berlin, New York: SpringerVerlag, ISBN 9780486687353, MR507446
Categories: Topology
 Topological spaces
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