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.

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 one-point 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

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