Closed extension topology

From Wikipedia, the free encyclopedia

The Closed extension topology is a topology created on the extension of a set by extending the closed sets. That is, if we take an already existing set Y and a point q, we create a new topology on Y* = Y ∪ {q} by simply defining the closed sets on Y* to be the closed sets on Y (with the exception of Y*).

Alternately we can perform the same process on a set using the subspace topology. If we let X be any topological space and p∈ X, we define a new topology as follows: Let S ⊂ X, then S is open whenever: p ∈ S and S = p ∪ S' where S' is open in the subspace topology on X\{p}. We note the closed sets on X\{p} are the same as those on X, which shows the equivalence of the two definitions.

If we replace the criterion that X\S be closed with the criterion that it be compact then we convert from the closed extension topology to the one-point compactification.

The closed extension topology shares many features in common with the more trivial particular point topology. In fact, if X\{p} is a set with a discrete topology then the closed extension topology is the particular point topology on X.

[edit] References

• Lynn Arthur Steen and J. Arthur Seebach, Jr., Counterexamples in Topology. Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995. ISBN 0-486-68735-X (Dover edition).