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.
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. As subsets of X ∪ P the subspace topology of X is the original topology of X, while the subspace topology of P is the discrete topology. As a topological space, X ∪ P is homeomorphic to the topological sum of X and P, and X is a clopen subset of X ∪ P.
If Y is a topological space and R is 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 similarity 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 , T ) {\displaystyle (X,{\mathcal {T}})} be a topological space and P {\displaystyle P} a set disjoint from X {\displaystyle X} . The open extension topology of T {\displaystyle {\mathcal {T}}} plus P {\displaystyle P} is T ∗ = T ∪ { X ∪ A : A ⊂ P } . {\displaystyle {\mathcal {T}}^{*}={\mathcal {T}}\cup \{X\cup A:A\subset P\}.} Let X ∗ = X ∪ P {\displaystyle X^{*}=X\cup P} . Then T ∗ {\displaystyle {\mathcal {T}}^{*}} is a topology in X ∗ {\displaystyle X^{*}} . The subspace topology of X {\displaystyle X} is the original topology of X {\displaystyle X} , i.e. T ∗ | X = T {\displaystyle {\mathcal {T}}^{*}|X={\mathcal {T}}} , while the subspace topology of P {\displaystyle P} is the discrete topology, i.e. T ∗ | P = P ( P ) {\displaystyle {\mathcal {T}}^{*}|P={\mathcal {P}}(P)} .
The closed sets in X ∗ {\displaystyle X^{*}} are { B ∪ P : X ⊂ B ∧ X ∖ B ∈ T } {\displaystyle \{B\cup P:X\subset B\land X\setminus B\in {\mathcal {T}}\}} . Note that P {\displaystyle P} is closed in X ∗ {\displaystyle X^{*}} and X {\displaystyle X} is open and dense in X ∗ {\displaystyle X^{*}} .
If Y a topological space and R is a subset of Y, one might ask whether the open 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 ∗ {\displaystyle X^{*}} is smaller than the extension topology of X ∗ {\displaystyle X^{*}} .
Assuming X {\displaystyle X} and P {\displaystyle P} are not empty to avoid trivialities, here are a few general properties of the open extension topology:1
- X {\displaystyle X} is dense in X ∗ {\displaystyle X^{*}} .
- If P {\displaystyle P} is finite, X ∗ {\displaystyle X^{*}} is compact. So X ∗ {\displaystyle X^{*}} is a compactification of X {\displaystyle X} in that case.
- X ∗ {\displaystyle X^{*}} is connected.
- If P {\displaystyle P} has a single point, X ∗ {\displaystyle X^{*}} is ultraconnected.
For a set Z and a point p 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. As subsets of X ∪ P the subspace topology of X is the original topology of X, while the subspace topology of P is the discrete topology.
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. Note that P is open in X ∪ P and X is closed in X ∪ P.
If Y is a topological space and R is a subset of Y, one might ask whether the closed 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.
For a set Z and a point p 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.
Notes
Works cited
- Steen, Lynn Arthur; Seebach, J. Arthur Jr (1995) [First published 1978], Counterexamples in Topology (Dover reprint of 1978 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-486-68735-3, MR 0507446
References
Steen & Seebach 1995, p. 48. - Steen, Lynn Arthur; Seebach, J. Arthur Jr (1995) [First published 1978], Counterexamples in Topology (Dover reprint of 1978 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-486-68735-3, MR 0507446 https://mathscinet.ams.org/mathscinet-getitem?mr=0507446 ↩