Extension Topology - Closed Extension Topology

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. 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 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.

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 closed 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 particular point topology construction by considering in Z the discrete topology and applying the closed extension topology construction to Z - {p} plus p.