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 ${\displaystyle (X,{\mathcal {T}})}$ be a topological space and ${\displaystyle P}$ a set disjoint from ${\displaystyle X}$. The open extension topology of ${\displaystyle {\mathcal {T}}}$ plus ${\displaystyle P}$ is $${\displaystyle {\mathcal {T}}^{*}={\mathcal {T}}\cup \{X\cup A:A\subset P\}.}$$Let ${\displaystyle X^{*}=X\cup P}$. Then ${\displaystyle {\mathcal {T}}^{*}}$ is a topology in ${\displaystyle X^{*}}$. The subspace topology of ${\displaystyle X}$ is the original topology of ${\displaystyle X}$, i.e. ${\displaystyle {\mathcal {T}}^{*}|X={\mathcal {T}}}$, while the subspace topology of ${\displaystyle P}$ is the discrete topology, i.e. ${\displaystyle {\mathcal {T}}^{*}|P={\mathcal {P}}(P)}$.

The closed sets in ${\displaystyle X^{*}}$ are ${\displaystyle \{B\cup P:X\subset B\land X\setminus B\in {\mathcal {T}}\}}$. Note that ${\displaystyle P}$ is closed in ${\displaystyle X^{*}}$ and ${\displaystyle X}$ is open and dense in ${\displaystyle X^{*}}$.

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 ${\displaystyle X^{*}}$ is smaller than the extension topology of ${\displaystyle X^{*}}$.

Assuming ${\displaystyle X}$ and ${\displaystyle P}$ are not empty to avoid trivialities, here are a few general properties of the open extension topology: ^[1]

• ${\displaystyle X}$ is dense in ${\displaystyle X^{*}}$.
• If ${\displaystyle P}$ is finite, ${\displaystyle X^{*}}$ is compact. So ${\displaystyle X^{*}}$ is a compactification of ${\displaystyle X}$ in that case.
• ${\displaystyle X^{*}}$ is connected.
• If ${\displaystyle P}$ has a single point, ${\displaystyle X^{*}}$ 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

1. Steen & Seebach 1995, p. 48.

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