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 YR 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 be a topological space and a set disjoint from . The open extension topology of plus is

Let . Then is a topology in . The subspace topology of is the original topology of , i.e. , while the subspace topology of is the discrete topology, i.e. .

The closed sets in are . Note that is closed in and is open and dense in .

If Y a topological space and R is a subset of Y, one might ask whether the open extension topology of YR 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 is smaller than the extension topology of .

Assuming and are not empty to avoid trivialities, here are a few general properties of the open extension topology:[1]

  • is dense in .
  • If is finite, is compact. So is a compactification of in that case.
  • is connected.
  • If has a single point, 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 YR 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
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