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:

    A breeze discovered my open book
    And began to flutter the leaves to look
    For a poem there used to be on Spring.
    Robert Frost (1874–1963)

    Slavery is founded on the selfishness of man’s nature—opposition to it on his love of justice. These principles are in eternal antagonism; and when brought into collision so fiercely as slavery extension brings them, shocks and throes and convulsions must ceaselessly follow.
    Abraham Lincoln (1809–1865)