Extension Topology - Open Extension Topology

Open 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: X ∪ Q, where Q is a subset of P, or A, where A is an open set of X.

For this reason this topology is called the open extension topology of X plus P, with which one extends to X ∪ P the open sets of X. Note that the subspace topology of X as a subset of X ∪ P is the original topology of X, while the subspace topology of P as a subset of X ∪ P is the discrete topology.

Note that the closed sets of X ∪ P are of the form: Q, where Q is a subset of P, or B ∪ P, where B is a closed set of X.

Being Y a topological space and R 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 that the open extension topology of X ∪ P is smaller than the extension topology of X ∪ P.

Being Z a set and p a point 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.

Read more about this topic:  Extension Topology

Famous quotes containing the words open and/or extension:

    Meanwhile Snow White held court,
    rolling her china-blue doll eyes open and shut
    and sometimes referring to her mirror
    as women do.
    Anne Sexton (1928–1974)

    Where there is reverence there is fear, but there is not reverence everywhere that there is fear, because fear presumably has a wider extension than reverence.
    Socrates (469–399 B.C.)