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.
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 [math]\displaystyle{ (X, \mathcal{T}) }[/math] be a topological space and [math]\displaystyle{ P }[/math] a set disjoint from [math]\displaystyle{ X }[/math]. The open extension topology of [math]\displaystyle{ \mathcal{T} }[/math] plus [math]\displaystyle{ P }[/math] is [math]\displaystyle{ \mathcal{T}^* = \mathcal{T} \cup \{X \cup A : A \subset P\}. }[/math]Let [math]\displaystyle{ X^* = X \cup P }[/math]. Then [math]\displaystyle{ \mathcal{T}^* }[/math]is a topology in [math]\displaystyle{ X^* }[/math]. The subspace topology of [math]\displaystyle{ X }[/math] is the original topology of [math]\displaystyle{ X }[/math], i.e. [math]\displaystyle{ \mathcal{T}^*|X = \mathcal{T} }[/math], while the subspace topology of [math]\displaystyle{ P }[/math] is the discrete topology, i.e. [math]\displaystyle{ \mathcal{T}^*|P = \mathcal{P}(P) }[/math].
The closed sets in [math]\displaystyle{ X^* }[/math] are [math]\displaystyle{ \{B \cup P : X \subset B \land X \setminus B \in \mathcal{T}\} }[/math]. Note that [math]\displaystyle{ P }[/math] is closed in [math]\displaystyle{ X^* }[/math] and [math]\displaystyle{ X }[/math] is open and dense in [math]\displaystyle{ X^* }[/math].
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 [math]\displaystyle{ X^* }[/math] is smaller than the extension topology of [math]\displaystyle{ X^* }[/math].
Assuming [math]\displaystyle{ X }[/math] and [math]\displaystyle{ P }[/math] are not empty to avoid trivialities, here are a few general properties of the open extension topology:[1]
- [math]\displaystyle{ X }[/math] is dense in [math]\displaystyle{ X^* }[/math].
- If [math]\displaystyle{ P }[/math] is finite, [math]\displaystyle{ X^* }[/math] is compact. So [math]\displaystyle{ X^* }[/math] is a compactification of [math]\displaystyle{ X }[/math] in that case.
- [math]\displaystyle{ X^* }[/math] is connected.
- If [math]\displaystyle{ P }[/math] has a single point, [math]\displaystyle{ X^* }[/math] 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
- ↑ Steen & Seebach 1995, p. 48.
Works cited
- Steen, Lynn Arthur; Seebach, J. Arthur Jr (1995), Counterexamples in Topology (Dover reprint of 1978 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-486-68735-3
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